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3.12.88 \(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{(d+e x)^3} \, dx\) [1188]

Optimal. Leaf size=508 \[ \frac {5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^6}-\frac {5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac {5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{3/2} e^7}+\frac {5 \sqrt {d} \sqrt {c d-b e} \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 e^7} \]

[Out]

-5/24*(B*d*(-13*b*e+24*c*d)-2*A*e*(-3*b*e+8*c*d)+e*(-4*A*c*e-B*b*e+6*B*c*d)*x)*(c*x^2+b*x)^(3/2)/e^4/(e*x+d)+1
/4*(B*e*x-2*A*e+3*B*d)*(c*x^2+b*x)^(5/2)/e^2/(e*x+d)^2-5/64*(8*A*c*e*(-b^3*e^3+18*b^2*c*d*e^2-48*b*c^2*d^2*e+3
2*c^3*d^3)-B*(-b^4*e^4-24*b^3*c*d*e^3+288*b^2*c^2*d^2*e^2-640*b*c^3*d^3*e+384*c^4*d^4))*arctanh(x*c^(1/2)/(c*x
^2+b*x)^(1/2))/c^(3/2)/e^7+5/8*(A*e*(3*b^2*e^2-16*b*c*d*e+16*c^2*d^2)-B*d*(7*b^2*e^2-28*b*c*d*e+24*c^2*d^2))*a
rctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))*d^(1/2)*(-b*e+c*d)^(1/2)/e^7+5/64*
(8*A*c*e*(5*b^2*e^2-20*b*c*d*e+16*c^2*d^2)-B*(-b^3*e^3+88*b^2*c*d*e^2-272*b*c^2*d^2*e+192*c^3*d^3)-2*c*e*(16*A
*c*e*(-b*e+2*c*d)-B*(b^2*e^2-32*b*c*d*e+48*c^2*d^2))*x)*(c*x^2+b*x)^(1/2)/c/e^6

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Rubi [A]
time = 0.48, antiderivative size = 508, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {826, 828, 857, 634, 212, 738} \begin {gather*} \frac {5 \sqrt {d} \sqrt {c d-b e} \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 e^7}+\frac {5 \sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-B \left (-b^3 e^3+88 b^2 c d e^2-272 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 c e^6}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{64 c^{3/2} e^7}-\frac {5 \left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{24 e^4 (d+e x)}+\frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(5*(8*A*c*e*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2) - B*(192*c^3*d^3 - 272*b*c^2*d^2*e + 88*b^2*c*d*e^2 - b^3*e^
3) - 2*c*e*(16*A*c*e*(2*c*d - b*e) - B*(48*c^2*d^2 - 32*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c*e^6) -
 (5*(B*d*(24*c*d - 13*b*e) - 2*A*e*(8*c*d - 3*b*e) + e*(6*B*c*d - b*B*e - 4*A*c*e)*x)*(b*x + c*x^2)^(3/2))/(24
*e^4*(d + e*x)) + ((3*B*d - 2*A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^2) - (5*(8*A*c*e*(32*c^3*d^3
- 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*e^3) - B*(384*c^4*d^4 - 640*b*c^3*d^3*e + 288*b^2*c^2*d^2*e^2 - 24*b^3
*c*d*e^3 - b^4*e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^7) + (5*Sqrt[d]*Sqrt[c*d - b*e]*(A*
e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) - B*d*(24*c^2*d^2 - 28*b*c*d*e + 7*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b
*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*e^7)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 826

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m +
 2*p + 2))), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
 + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
  !ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
 + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
 c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx &=\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac {5 \int \frac {(2 b (3 B d-2 A e)+2 (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx}{16 e^2}\\ &=-\frac {5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}+\frac {5 \int \frac {\left (2 b (B d (24 c d-13 b e)-2 A e (8 c d-3 b e))-2 \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{d+e x} \, dx}{32 e^4}\\ &=\frac {5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^6}-\frac {5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac {5 \int \frac {b d \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )\right )+\left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{128 c e^6}\\ &=\frac {5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^6}-\frac {5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}+\frac {\left (5 d (c d-b e) \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 e^7}-\frac {\left (5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c e^7}\\ &=\frac {5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^6}-\frac {5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac {\left (5 d (c d-b e) \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{4 e^7}-\frac {\left (5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{64 c e^7}\\ &=\frac {5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^6}-\frac {5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac {5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{3/2} e^7}+\frac {5 \sqrt {d} \sqrt {c d-b e} \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 e^7}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2047\) vs. \(2(508)=1016\).
time = 16.13, size = 2047, normalized size = 4.03 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(5/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((x*(b + c*x))^(5/2)*(((-5
*c*d*(B*d - A*e) + (e*(7*b*B*d - 4*A*c*d - 3*A*b*e))/2)*x^(7/2)*(b + c*x)^(7/2))/(d*(-(c*d) + b*e)*(d + e*x))
+ (((8*A*c^2*d^2 + 4*b*c*d*(14*B*d - 11*A*e) - 5*b^2*e*(7*B*d - 3*A*e))*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*
x)/b)^3*((5/(16*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (15*b^3*((2*c*x)/b - (4*c^2
*x^2)/(3*b^2) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(512*c^3*
x^3*(1 + (c*x)/b)^3)))/(5*e) - (d*((2*b^2*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b)^3) + 5
/(6*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/8 + (15*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[
x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(256*c^2*x^2*(1 + (c*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]*Sqrt[b
+ c*x]*(1 + (c*x)/b)^3*((15/(8*(1 + (c*x)/b)^3) + 5/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*Sqrt[b]*A
rcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(16*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2))))/e - (d*((2*b*c*Sqrt[x]*Sqrt[b +
c*x]*(1 + (c*x)/b)^2*((3/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sq
rt[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(5/2))))/e - ((c*d - b*e)*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)*(
1/(2*(1 + (c*x)/b)) + (Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2))))/e
 - ((c*d - b*e)*((2*Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b + c*x]) -
(2*Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e)))/e))/e))/e))/e))/e
))/4 + 3*c*(B*d*(10*c*d - 7*b*e) - 3*A*e*(2*c*d - b*e))*((2*b^2*x^(7/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((7*(3/(
16*(1 + (c*x)/b)^3) + 1/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/12 + (35*b^4*((2*c*x)/b - (4*c^2*x^2)/(3*b^
2) + (16*c^3*x^3)/(15*b^3) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]
)))/(2048*c^4*x^4*(1 + (c*x)/b)^3)))/(7*e) - (d*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((5/(16*(1 + (c*
x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (15*b^3*((2*c*x)/b - (4*c^2*x^2)/(3*b^2) - (2*Sqrt[
c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(512*c^3*x^3*(1 + (c*x)/b)^3)))/(
5*e) - (d*((2*b^2*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b)^3) + 5/(6*(1 + (c*x)/b)^2) + (
1 + (c*x)/b)^(-1)))/8 + (15*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*S
qrt[1 + (c*x)/b])))/(256*c^2*x^2*(1 + (c*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)^3*(
(15/(8*(1 + (c*x)/b)^3) + 5/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])
/Sqrt[b]])/(16*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2))))/e - (d*((2*b*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)^2*((3
/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(8*Sqrt[c]*Sqrt[
x]*(1 + (c*x)/b)^(5/2))))/e - ((c*d - b*e)*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)*(1/(2*(1 + (c*x)/b)) + (S
qrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2))))/e - ((c*d - b*e)*((2*Sqrt
[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b + c*x]) - (2*Sqrt[c*d - b*e]*ArcTa
nh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e)))/e))/e))/e))/e))/e))/e))/(d*(-(c*d) + b*e)
)))/(2*d*(-(c*d) + b*e)*x^(5/2)*(b + c*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3957\) vs. \(2(474)=948\).
time = 0.68, size = 3958, normalized size = 7.79

method result size
default \(\text {Expression too large to display}\) \(3958\)
risch \(\text {Expression too large to display}\) \(6960\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

B/e^3*(1/d/(b*e-c*d)*e^2/(x+d/e)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(7/2)-5/2*e*(b*e-2*c*d)
/d/(b*e-c*d)*(1/5*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(5/2)+1/2/e*(b*e-2*c*d)*(1/8*(2*c*(x+d
/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+3/16*(-4*c*d*(b*e-c*d)/e^2-
1/e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e
^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(
x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))))-d*(b*e-c*d)/e^2*(1/3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(
x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/2/e*(b*e-2*c*d)*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c
*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d
)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)))-d*(b*e-c*d)/e^2*((c*(x+d/e)
^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(
c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)+d*(b*e-c*d)/e^2/(-d*(b*e-c*d)/e^2)^(1/2)*l
n((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-
d*(b*e-c*d)/e^2)^(1/2))/(x+d/e)))))-6*c/d/(b*e-c*d)*e^2*(1/12*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e
*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(5/2)+5/24*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/8*(2*c*(x+d/e
)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+3/16*(-4*c*d*(b*e-c*d)/e^2-1/
e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2
)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+
d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))))))+(A*e-B*d)/e^4*(1/2/d/(b*e-c*d)*e^2/(x+d/e)^2*(c*(x+
d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(7/2)-3/4*e*(b*e-2*c*d)/d/(b*e-c*d)*(1/d/(b*e-c*d)*e^2/(x+d/e)
*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(7/2)-5/2*e*(b*e-2*c*d)/d/(b*e-c*d)*(1/5*(c*(x+d/e)^2+1
/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(5/2)+1/2/e*(b*e-2*c*d)*(1/8*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e
)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+3/16*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c
*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/
e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)
-d*(b*e-c*d)/e^2)^(1/2))))-d*(b*e-c*d)/e^2*(1/3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/
2/e*(b*e-2*c*d)*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/
2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^
2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)))-d*(b*e-c*d)/e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*
e-c*d)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d
/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)+d*(b*e-c*d)/e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2
*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))
)))-6*c/d/(b*e-c*d)*e^2*(1/12*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)
/e^2)^(5/2)+5/24*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/8*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^
2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+3/16*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c*(
x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^
2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d
*(b*e-c*d)/e^2)^(1/2))))))-5/2*c/d/(b*e-c*d)*e^2*(1/5*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(5
/2)+1/2/e*(b*e-2*c*d)*(1/8*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^
2)^(3/2)+3/16*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c*(1/4*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/c*(c*(x+d/e)^2+1
/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/c^(3/2)*ln((1/2/e
*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))))-d*(b*e-c*d)/e^2
*(1/3*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/2/e*(b*e-2*c*d)*(1/4*(2*c*(x+d/e)+1/e*(b*e
-2*c*d))/c*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/8*(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*
c*d)^2)/c^(3/2)*ln((1/2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)
^(1/2)))-d*(b*e-c*d)/e^2*((c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/2/e*(b*e-2*c*d)*ln((1/
2/e*(b*e-2*c*d)+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2...

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-%e*b>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (484) = 968\).
time = 27.97, size = 4007, normalized size = 7.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[-1/384*(15*(384*B*c^4*d^6 - (B*b^4 - 8*A*b^3*c)*x^2*e^6 - 2*(12*(B*b^3*c + 6*A*b^2*c^2)*d*x^2 + (B*b^4 - 8*A*
b^3*c)*d*x)*e^5 + (96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*x^2 - 48*(B*b^3*c + 6*A*b^2*c^2)*d^2*x - (B*b^4 - 8*A*b^3*
c)*d^2)*e^4 - 8*(16*(5*B*b*c^3 + 2*A*c^4)*d^3*x^2 - 24*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*x + 3*(B*b^3*c + 6*A*b^2*
c^2)*d^3)*e^3 + 32*(12*B*c^4*d^4*x^2 - 8*(5*B*b*c^3 + 2*A*c^4)*d^4*x + 3*(3*B*b^2*c^2 + 4*A*b*c^3)*d^4)*e^2 +
128*(6*B*c^4*d^5*x - (5*B*b*c^3 + 2*A*c^4)*d^5)*e)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 240*
(24*B*c^4*d^5 - 3*A*b^2*c^2*x^2*e^5 - (6*A*b^2*c^2*d*x - (7*B*b^2*c^2 + 16*A*b*c^3)*d*x^2)*e^4 - (3*A*b^2*c^2*
d^2 + 4*(7*B*b*c^3 + 4*A*c^4)*d^2*x^2 - 2*(7*B*b^2*c^2 + 16*A*b*c^3)*d^2*x)*e^3 + (24*B*c^4*d^3*x^2 - 8*(7*B*b
*c^3 + 4*A*c^4)*d^3*x + (7*B*b^2*c^2 + 16*A*b*c^3)*d^3)*e^2 + 4*(12*B*c^4*d^4*x - (7*B*b*c^3 + 4*A*c^4)*d^4)*e
)*sqrt(c*d^2 - b*d*e)*log((2*c*d*x - b*x*e + b*d - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(x*e + d)) + 2*(28
80*B*c^4*d^5*e - (48*B*c^4*x^5 + 8*(17*B*b*c^3 + 8*A*c^4)*x^4 + 2*(59*B*b^2*c^2 + 104*A*b*c^3)*x^3 + 3*(5*B*b^
3*c + 88*A*b^2*c^2)*x^2)*e^6 + 2*(48*B*c^4*d*x^4 + 16*(11*B*b*c^3 + 5*A*c^4)*d*x^3 + 2*(139*B*b^2*c^2 + 220*A*
b*c^3)*d*x^2 - 15*(B*b^3*c + 32*A*b^2*c^2)*d*x)*e^5 - 5*(48*B*c^4*d^2*x^3 + 8*(37*B*b*c^3 + 16*A*c^4)*d^2*x^2
- 2*(209*B*b^2*c^2 + 368*A*b*c^3)*d^2*x + 3*(B*b^3*c + 40*A*b^2*c^2)*d^2)*e^4 + 120*(8*B*c^4*d^3*x^2 - 4*(13*B
*b*c^3 + 6*A*c^4)*d^3*x + (11*B*b^2*c^2 + 20*A*b*c^3)*d^3)*e^3 + 240*(18*B*c^4*d^4*x - (17*B*b*c^3 + 8*A*c^4)*
d^4)*e^2)*sqrt(c*x^2 + b*x))/(c^2*x^2*e^9 + 2*c^2*d*x*e^8 + c^2*d^2*e^7), -1/384*(480*(24*B*c^4*d^5 - 3*A*b^2*
c^2*x^2*e^5 - (6*A*b^2*c^2*d*x - (7*B*b^2*c^2 + 16*A*b*c^3)*d*x^2)*e^4 - (3*A*b^2*c^2*d^2 + 4*(7*B*b*c^3 + 4*A
*c^4)*d^2*x^2 - 2*(7*B*b^2*c^2 + 16*A*b*c^3)*d^2*x)*e^3 + (24*B*c^4*d^3*x^2 - 8*(7*B*b*c^3 + 4*A*c^4)*d^3*x +
(7*B*b^2*c^2 + 16*A*b*c^3)*d^3)*e^2 + 4*(12*B*c^4*d^4*x - (7*B*b*c^3 + 4*A*c^4)*d^4)*e)*sqrt(-c*d^2 + b*d*e)*a
rctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e)) + 15*(384*B*c^4*d^6 - (B*b^4 - 8*A*b^3*c)*x^2*e
^6 - 2*(12*(B*b^3*c + 6*A*b^2*c^2)*d*x^2 + (B*b^4 - 8*A*b^3*c)*d*x)*e^5 + (96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*x^
2 - 48*(B*b^3*c + 6*A*b^2*c^2)*d^2*x - (B*b^4 - 8*A*b^3*c)*d^2)*e^4 - 8*(16*(5*B*b*c^3 + 2*A*c^4)*d^3*x^2 - 24
*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*x + 3*(B*b^3*c + 6*A*b^2*c^2)*d^3)*e^3 + 32*(12*B*c^4*d^4*x^2 - 8*(5*B*b*c^3 +
2*A*c^4)*d^4*x + 3*(3*B*b^2*c^2 + 4*A*b*c^3)*d^4)*e^2 + 128*(6*B*c^4*d^5*x - (5*B*b*c^3 + 2*A*c^4)*d^5)*e)*sqr
t(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(2880*B*c^4*d^5*e - (48*B*c^4*x^5 + 8*(17*B*b*c^3 + 8*A*
c^4)*x^4 + 2*(59*B*b^2*c^2 + 104*A*b*c^3)*x^3 + 3*(5*B*b^3*c + 88*A*b^2*c^2)*x^2)*e^6 + 2*(48*B*c^4*d*x^4 + 16
*(11*B*b*c^3 + 5*A*c^4)*d*x^3 + 2*(139*B*b^2*c^2 + 220*A*b*c^3)*d*x^2 - 15*(B*b^3*c + 32*A*b^2*c^2)*d*x)*e^5 -
 5*(48*B*c^4*d^2*x^3 + 8*(37*B*b*c^3 + 16*A*c^4)*d^2*x^2 - 2*(209*B*b^2*c^2 + 368*A*b*c^3)*d^2*x + 3*(B*b^3*c
+ 40*A*b^2*c^2)*d^2)*e^4 + 120*(8*B*c^4*d^3*x^2 - 4*(13*B*b*c^3 + 6*A*c^4)*d^3*x + (11*B*b^2*c^2 + 20*A*b*c^3)
*d^3)*e^3 + 240*(18*B*c^4*d^4*x - (17*B*b*c^3 + 8*A*c^4)*d^4)*e^2)*sqrt(c*x^2 + b*x))/(c^2*x^2*e^9 + 2*c^2*d*x
*e^8 + c^2*d^2*e^7), -1/192*(15*(384*B*c^4*d^6 - (B*b^4 - 8*A*b^3*c)*x^2*e^6 - 2*(12*(B*b^3*c + 6*A*b^2*c^2)*d
*x^2 + (B*b^4 - 8*A*b^3*c)*d*x)*e^5 + (96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*x^2 - 48*(B*b^3*c + 6*A*b^2*c^2)*d^2*x
 - (B*b^4 - 8*A*b^3*c)*d^2)*e^4 - 8*(16*(5*B*b*c^3 + 2*A*c^4)*d^3*x^2 - 24*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*x + 3
*(B*b^3*c + 6*A*b^2*c^2)*d^3)*e^3 + 32*(12*B*c^4*d^4*x^2 - 8*(5*B*b*c^3 + 2*A*c^4)*d^4*x + 3*(3*B*b^2*c^2 + 4*
A*b*c^3)*d^4)*e^2 + 128*(6*B*c^4*d^5*x - (5*B*b*c^3 + 2*A*c^4)*d^5)*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(
-c)/(c*x)) - 120*(24*B*c^4*d^5 - 3*A*b^2*c^2*x^2*e^5 - (6*A*b^2*c^2*d*x - (7*B*b^2*c^2 + 16*A*b*c^3)*d*x^2)*e^
4 - (3*A*b^2*c^2*d^2 + 4*(7*B*b*c^3 + 4*A*c^4)*d^2*x^2 - 2*(7*B*b^2*c^2 + 16*A*b*c^3)*d^2*x)*e^3 + (24*B*c^4*d
^3*x^2 - 8*(7*B*b*c^3 + 4*A*c^4)*d^3*x + (7*B*b^2*c^2 + 16*A*b*c^3)*d^3)*e^2 + 4*(12*B*c^4*d^4*x - (7*B*b*c^3
+ 4*A*c^4)*d^4)*e)*sqrt(c*d^2 - b*d*e)*log((2*c*d*x - b*x*e + b*d - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(
x*e + d)) + (2880*B*c^4*d^5*e - (48*B*c^4*x^5 + 8*(17*B*b*c^3 + 8*A*c^4)*x^4 + 2*(59*B*b^2*c^2 + 104*A*b*c^3)*
x^3 + 3*(5*B*b^3*c + 88*A*b^2*c^2)*x^2)*e^6 + 2*(48*B*c^4*d*x^4 + 16*(11*B*b*c^3 + 5*A*c^4)*d*x^3 + 2*(139*B*b
^2*c^2 + 220*A*b*c^3)*d*x^2 - 15*(B*b^3*c + 32*A*b^2*c^2)*d*x)*e^5 - 5*(48*B*c^4*d^2*x^3 + 8*(37*B*b*c^3 + 16*
A*c^4)*d^2*x^2 - 2*(209*B*b^2*c^2 + 368*A*b*c^3)*d^2*x + 3*(B*b^3*c + 40*A*b^2*c^2)*d^2)*e^4 + 120*(8*B*c^4*d^
3*x^2 - 4*(13*B*b*c^3 + 6*A*c^4)*d^3*x + (11*B*b^2*c^2 + 20*A*b*c^3)*d^3)*e^3 + 240*(18*B*c^4*d^4*x - (17*B*b*
c^3 + 8*A*c^4)*d^4)*e^2)*sqrt(c*x^2 + b*x))/(c^2*x^2*e^9 + 2*c^2*d*x*e^8 + c^2*d^2*e^7), -1/192*(240*(24*B*c^4
*d^5 - 3*A*b^2*c^2*x^2*e^5 - (6*A*b^2*c^2*d*x - (7*B*b^2*c^2 + 16*A*b*c^3)*d*x^2)*e^4 - (3*A*b^2*c^2*d^2 + 4*(
7*B*b*c^3 + 4*A*c^4)*d^2*x^2 - 2*(7*B*b^2*c^2 +...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(5/2)/(e*x+d)**3,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/(d + e*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1425 vs. \(2 (484) = 968\).
time = 4.88, size = 1425, normalized size = 2.81 \begin {gather*} -\frac {5 \, {\left (24 \, B c^{3} d^{5} - 52 \, B b c^{2} d^{4} e - 16 \, A c^{3} d^{4} e + 35 \, B b^{2} c d^{3} e^{2} + 32 \, A b c^{2} d^{3} e^{2} - 7 \, B b^{3} d^{2} e^{3} - 19 \, A b^{2} c d^{2} e^{3} + 3 \, A b^{3} d e^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right ) e^{\left (-7\right )}}{4 \, \sqrt {-c d^{2} + b d e}} + \frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, B c^{2} x e^{\left (-3\right )} - \frac {{\left (24 \, B c^{5} d e^{21} - 17 \, B b c^{4} e^{22} - 8 \, A c^{5} e^{22}\right )} e^{\left (-25\right )}}{c^{3}}\right )} x + \frac {{\left (288 \, B c^{5} d^{2} e^{20} - 312 \, B b c^{4} d e^{21} - 144 \, A c^{5} d e^{21} + 59 \, B b^{2} c^{3} e^{22} + 104 \, A b c^{4} e^{22}\right )} e^{\left (-25\right )}}{c^{3}}\right )} x - \frac {3 \, {\left (640 \, B c^{5} d^{3} e^{19} - 864 \, B b c^{4} d^{2} e^{20} - 384 \, A c^{5} d^{2} e^{20} + 264 \, B b^{2} c^{3} d e^{21} + 432 \, A b c^{4} d e^{21} - 5 \, B b^{3} c^{2} e^{22} - 88 \, A b^{2} c^{3} e^{22}\right )} e^{\left (-25\right )}}{c^{3}}\right )} - \frac {5 \, {\left (384 \, B c^{4} d^{4} - 640 \, B b c^{3} d^{3} e - 256 \, A c^{4} d^{3} e + 288 \, B b^{2} c^{2} d^{2} e^{2} + 384 \, A b c^{3} d^{2} e^{2} - 24 \, B b^{3} c d e^{3} - 144 \, A b^{2} c^{2} d e^{3} - B b^{4} e^{4} + 8 \, A b^{3} c e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {3}{2}}} - \frac {{\left (48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B c^{\frac {7}{2}} d^{5} e + 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c^{4} d^{6} - 156 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b c^{3} d^{5} e - 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c^{4} d^{5} e + 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b c^{\frac {7}{2}} d^{6} - 100 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b c^{\frac {5}{2}} d^{4} e^{2} - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{\frac {7}{2}} d^{4} e^{2} - 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{2} c^{\frac {5}{2}} d^{5} e - 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b c^{\frac {7}{2}} d^{5} e + 22 \, B b^{2} c^{3} d^{6} + 75 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} c^{2} d^{4} e^{2} + 120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c^{3} d^{4} e^{2} - 35 \, B b^{3} c^{2} d^{5} e - 18 \, A b^{2} c^{3} d^{5} e + 65 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} c^{\frac {3}{2}} d^{3} e^{3} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{\frac {5}{2}} d^{3} e^{3} + 83 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} c^{\frac {3}{2}} d^{4} e^{2} + 124 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c^{\frac {5}{2}} d^{4} e^{2} - 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{3} c d^{3} e^{3} - 51 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c^{2} d^{3} e^{3} + 13 \, B b^{4} c d^{4} e^{2} + 27 \, A b^{3} c^{2} d^{4} e^{2} - 13 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{3} \sqrt {c} d^{2} e^{4} - 49 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac {3}{2}} d^{2} e^{4} - 11 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{4} \sqrt {c} d^{3} e^{3} - 59 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} c^{\frac {3}{2}} d^{3} e^{3} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{3} c d^{2} e^{4} - 9 \, A b^{4} c d^{3} e^{3} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{3} \sqrt {c} d e^{5} + 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{4} \sqrt {c} d^{2} e^{4}\right )} e^{\left (-7\right )}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

-5/4*(24*B*c^3*d^5 - 52*B*b*c^2*d^4*e - 16*A*c^3*d^4*e + 35*B*b^2*c*d^3*e^2 + 32*A*b*c^2*d^3*e^2 - 7*B*b^3*d^2
*e^3 - 19*A*b^2*c*d^2*e^3 + 3*A*b^3*d*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2
 + b*d*e))*e^(-7)/sqrt(-c*d^2 + b*d*e) + 1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*c^2*x*e^(-3) - (24*B*c^5*d*e^21 -
17*B*b*c^4*e^22 - 8*A*c^5*e^22)*e^(-25)/c^3)*x + (288*B*c^5*d^2*e^20 - 312*B*b*c^4*d*e^21 - 144*A*c^5*d*e^21 +
 59*B*b^2*c^3*e^22 + 104*A*b*c^4*e^22)*e^(-25)/c^3)*x - 3*(640*B*c^5*d^3*e^19 - 864*B*b*c^4*d^2*e^20 - 384*A*c
^5*d^2*e^20 + 264*B*b^2*c^3*d*e^21 + 432*A*b*c^4*d*e^21 - 5*B*b^3*c^2*e^22 - 88*A*b^2*c^3*e^22)*e^(-25)/c^3) -
 5/128*(384*B*c^4*d^4 - 640*B*b*c^3*d^3*e - 256*A*c^4*d^3*e + 288*B*b^2*c^2*d^2*e^2 + 384*A*b*c^3*d^2*e^2 - 24
*B*b^3*c*d*e^3 - 144*A*b^2*c^2*d*e^3 - B*b^4*e^4 + 8*A*b^3*c*e^4)*e^(-7)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b
*x))*sqrt(c) + b))/c^(3/2) - 1/4*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^(7/2)*d^5*e + 88*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^2*B*c^4*d^6 - 156*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^3*d^5*e - 72*(sqrt(c)*x - sqrt(c*x^2 +
 b*x))^2*A*c^4*d^5*e + 88*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^(7/2)*d^6 - 100*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^3*B*b*c^(5/2)*d^4*e^2 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(7/2)*d^4*e^2 - 160*(sqrt(c)*x - sqrt(c*x^2
 + b*x))*B*b^2*c^(5/2)*d^5*e - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^(7/2)*d^5*e + 22*B*b^2*c^3*d^6 + 75*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*c^2*d^4*e^2 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^3*d^4*e^2 - 35
*B*b^3*c^2*d^5*e - 18*A*b^2*c^3*d^5*e + 65*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c^(3/2)*d^3*e^3 + 80*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^(5/2)*d^3*e^3 + 83*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*c^(3/2)*d^4*e^2 + 1
24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c^(5/2)*d^4*e^2 - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*c*d^3*e^3
 - 51*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c^2*d^3*e^3 + 13*B*b^4*c*d^4*e^2 + 27*A*b^3*c^2*d^4*e^2 - 13*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*sqrt(c)*d^2*e^4 - 49*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(3/2)*d^2*
e^4 - 11*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*sqrt(c)*d^3*e^3 - 59*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*c^(3
/2)*d^3*e^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^3*c*d^2*e^4 - 9*A*b^4*c*d^3*e^3 + 9*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^3*A*b^3*sqrt(c)*d*e^5 + 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*sqrt(c)*d^2*e^4)*e^(-7)/(((sqrt(c)
*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2*sqrt(c))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^(5/2)*(A + B*x))/(d + e*x)^3,x)

[Out]

int(((b*x + c*x^2)^(5/2)*(A + B*x))/(d + e*x)^3, x)

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