3.11.31 \(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=323 \[ -\frac {\sqrt {b x+c x^2} \left (2 c e x (-6 A c e-b B e+8 B c d)+6 A c e (4 c d-3 b e)-B \left (b^2 e^2-28 b c d e+32 c^2 d^2\right )\right )}{8 c e^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 b c e (4 B d-3 A e) (2 c d-b e)-\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) (-6 A c e-b B e+8 B c d)\right )}{8 c^{3/2} e^5}+\frac {\sqrt {d} \sqrt {c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 e^5}+\frac {\left (b x+c x^2\right )^{3/2} (-3 A e+4 B d+B e x)}{3 e^2 (d+e x)} \]

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Rubi [A]  time = 0.42, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {812, 814, 843, 620, 206, 724} \begin {gather*} -\frac {\sqrt {b x+c x^2} \left (2 c e x (-6 A c e-b B e+8 B c d)+6 A c e (4 c d-3 b e)-B \left (b^2 e^2-28 b c d e+32 c^2 d^2\right )\right )}{8 c e^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 b c e (4 B d-3 A e) (2 c d-b e)-\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) (-6 A c e-b B e+8 B c d)\right )}{8 c^{3/2} e^5}+\frac {\left (b x+c x^2\right )^{3/2} (-3 A e+4 B d+B e x)}{3 e^2 (d+e x)}+\frac {\sqrt {d} \sqrt {c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \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 )}{2 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((6*A*c*e*(4*c*d - 3*b*e) - B*(32*c^2*d^2 - 28*b*c*d*e + b^2*e^2) + 2*c*e*(8*B*c*d - b*B*e - 6*A*c*e)*x)*Sqrt
[b*x + c*x^2])/(8*c*e^4) + ((4*B*d - 3*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) + ((4*b*c*e*(4*B*d
- 3*A*e)*(2*c*d - b*e) - (8*B*c*d - b*B*e - 6*A*c*e)*(8*c^2*d^2 - 4*b*c*d*e - b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sq
rt[b*x + c*x^2]])/(8*c^(3/2)*e^5) + (Sqrt[d]*Sqrt[c*d - b*e]*(B*d*(8*c*d - 5*b*e) - 3*A*e*(2*c*d - b*e))*ArcTa
nh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^5)

Rule 206

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

Rule 620

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 724

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 812

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 814

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 843

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 )^{3/2}}{(d+e x)^2} \, dx &=\frac {(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {\int \frac {(b (4 B d-3 A e)+(8 B c d-b B e-6 A c e) x) \sqrt {b x+c x^2}}{d+e x} \, dx}{2 e^2}\\ &=-\frac {\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt {b x+c x^2}}{8 c e^4}+\frac {(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}+\frac {\int \frac {\frac {1}{2} b d \left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )\right )+\frac {1}{2} \left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 c e^4}\\ &=-\frac {\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt {b x+c x^2}}{8 c e^4}+\frac {(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}+\frac {(d (c d-b e) (B d (8 c d-5 b e)-3 A e (2 c d-b e))) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 e^5}+\frac {\left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c e^5}\\ &=-\frac {\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt {b x+c x^2}}{8 c e^4}+\frac {(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {(d (c d-b e) (B d (8 c d-5 b e)-3 A e (2 c d-b e))) \operatorname {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 )}{e^5}+\frac {\left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c e^5}\\ &=-\frac {\left (6 A c e (4 c d-3 b e)-B \left (32 c^2 d^2-28 b c d e+b^2 e^2\right )+2 c e (8 B c d-b B e-6 A c e) x\right ) \sqrt {b x+c x^2}}{8 c e^4}+\frac {(4 B d-3 A e+B e x) \left (b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}+\frac {\left (4 b c e (4 B d-3 A e) (2 c d-b e)-(8 B c d-b B e-6 A c e) \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^5}+\frac {\sqrt {d} \sqrt {c d-b e} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \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 )}{2 e^5}\\ \end {align*}

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Mathematica [A]  time = 1.64, size = 353, normalized size = 1.09 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (\frac {e \sqrt {x} \left (6 A c e \left (b e (9 d+5 e x)-2 c \left (6 d^2+3 d e x-e^2 x^2\right )\right )+B \left (3 b^2 e^2 (d+e x)+2 b c e \left (-42 d^2-23 d e x+7 e^2 x^2\right )+8 c^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{d+e x}+\frac {24 c \sqrt {d} \sqrt {c d-b e} (3 A e (b e-2 c d)+B d (8 c d-5 b e)) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )-\frac {3 \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (B \left (b^3 e^3+12 b^2 c d e^2-72 b c^2 d^2 e+64 c^3 d^3\right )-6 A c e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )\right )}{\sqrt {b} \sqrt {\frac {c x}{b}+1}}\right )}{24 c^{3/2} e^5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*((-3*(-6*A*c*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) + B*(64*c^3*d^3 - 72*b*c^2*d^2*e + 12*b^2*
c*d*e^2 + b^3*e^3))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) + Sqrt[c]*((e*Sqrt[x]*(6*A
*c*e*(b*e*(9*d + 5*e*x) - 2*c*(6*d^2 + 3*d*e*x - e^2*x^2)) + B*(3*b^2*e^2*(d + e*x) + 2*b*c*e*(-42*d^2 - 23*d*
e*x + 7*e^2*x^2) + 8*c^2*(12*d^3 + 6*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3))))/(d + e*x) + (24*c*Sqrt[d]*Sqrt[c*d -
b*e]*(B*d*(8*c*d - 5*b*e) + 3*A*e*(-2*c*d + b*e))*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/
Sqrt[b + c*x])))/(24*c^(3/2)*e^5*Sqrt[x])

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IntegrateAlgebraic [A]  time = 2.95, size = 437, normalized size = 1.35 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (54 A b c d e^2+30 A b c e^3 x-72 A c^2 d^2 e-36 A c^2 d e^2 x+12 A c^2 e^3 x^2+3 b^2 B d e^2+3 b^2 B e^3 x-84 b B c d^2 e-46 b B c d e^2 x+14 b B c e^3 x^2+96 B c^2 d^3+48 B c^2 d^2 e x-16 B c^2 d e^2 x^2+8 B c^2 e^3 x^3\right )}{24 c e^4 (d+e x)}+\frac {\log \left (-2 c^{3/2} \sqrt {b x+c x^2}+b c+2 c^2 x\right ) \left (-6 A b^2 c e^3+48 A b c^2 d e^2-48 A c^3 d^2 e+b^3 B e^3+12 b^2 B c d e^2-72 b B c^2 d^2 e+64 B c^3 d^3\right )}{16 c^{3/2} e^5}+\frac {\left (-6 A c d^{3/2} e \sqrt {c d-b e}+3 A b \sqrt {d} e^2 \sqrt {c d-b e}+8 B c d^{5/2} \sqrt {c d-b e}-5 b B d^{3/2} e \sqrt {c d-b e}\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(96*B*c^2*d^3 - 84*b*B*c*d^2*e - 72*A*c^2*d^2*e + 3*b^2*B*d*e^2 + 54*A*b*c*d*e^2 + 48*B*c^2
*d^2*e*x - 46*b*B*c*d*e^2*x - 36*A*c^2*d*e^2*x + 3*b^2*B*e^3*x + 30*A*b*c*e^3*x - 16*B*c^2*d*e^2*x^2 + 14*b*B*
c*e^3*x^2 + 12*A*c^2*e^3*x^2 + 8*B*c^2*e^3*x^3))/(24*c*e^4*(d + e*x)) + ((8*B*c*d^(5/2)*Sqrt[c*d - b*e] - 5*b*
B*d^(3/2)*e*Sqrt[c*d - b*e] - 6*A*c*d^(3/2)*e*Sqrt[c*d - b*e] + 3*A*b*Sqrt[d]*e^2*Sqrt[c*d - b*e])*ArcTanh[(Sq
rt[c]*d + Sqrt[c]*e*x - e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/e^5 + ((64*B*c^3*d^3 - 72*b*B*c^2*d^2
*e - 48*A*c^3*d^2*e + 12*b^2*B*c*d*e^2 + 48*A*b*c^2*d*e^2 + b^3*B*e^3 - 6*A*b^2*c*e^3)*Log[b*c + 2*c^2*x - 2*c
^(3/2)*Sqrt[b*x + c*x^2]])/(16*c^(3/2)*e^5)

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fricas [A]  time = 3.86, size = 2017, normalized size = 6.24

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2
*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A
*b^2*c)*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 24*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*
B*b*c^2 + 6*A*c^3)*d^2*e + (8*B*c^3*d^2*e + 3*A*b*c^2*e^3 - (5*B*b*c^2 + 6*A*c^3)*d*e^2)*x)*sqrt(c*d^2 - b*d*e
)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(8*B*c^3*e^4*x^3 + 96*B
*c^3*d^3*e - 12*(7*B*b*c^2 + 6*A*c^3)*d^2*e^2 + 3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(8*B*c^3*d*e^3 - (7*B*b*c^2
+ 6*A*c^3)*e^4)*x^2 + (48*B*c^3*d^2*e^2 - 2*(23*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^2*c + 10*A*b*c^2)*e^4)*x)*
sqrt(c*x^2 + b*x))/(c^2*e^6*x + c^2*d*e^5), 1/48*(48*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*B*b*c^2 + 6*A*c^3)*d^
2*e + (8*B*c^3*d^2*e + 3*A*b*c^2*e^3 - (5*B*b*c^2 + 6*A*c^3)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^
2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c
+ 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B
*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) +
2*(8*B*c^3*e^4*x^3 + 96*B*c^3*d^3*e - 12*(7*B*b*c^2 + 6*A*c^3)*d^2*e^2 + 3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(
8*B*c^3*d*e^3 - (7*B*b*c^2 + 6*A*c^3)*e^4)*x^2 + (48*B*c^3*d^2*e^2 - 2*(23*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^
2*c + 10*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*e^6*x + c^2*d*e^5), 1/24*(3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 +
2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^
2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2
+ b*x)*sqrt(-c)/(c*x)) + 12*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*B*b*c^2 + 6*A*c^3)*d^2*e + (8*B*c^3*d^2*e + 3
*A*b*c^2*e^3 - (5*B*b*c^2 + 6*A*c^3)*d*e^2)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 -
b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + (8*B*c^3*e^4*x^3 + 96*B*c^3*d^3*e - 12*(7*B*b*c^2 + 6*A*c^3)*d^2*e^2 +
3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(8*B*c^3*d*e^3 - (7*B*b*c^2 + 6*A*c^3)*e^4)*x^2 + (48*B*c^3*d^2*e^2 - 2*(2
3*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^2*c + 10*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*e^6*x + c^2*d*e^5), 1/2
4*(24*(8*B*c^3*d^3 + 3*A*b*c^2*d*e^2 - (5*B*b*c^2 + 6*A*c^3)*d^2*e + (8*B*c^3*d^2*e + 3*A*b*c^2*e^3 - (5*B*b*c
^2 + 6*A*c^3)*d*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) +
3*(64*B*c^3*d^4 - 24*(3*B*b*c^2 + 2*A*c^3)*d^3*e + 12*(B*b^2*c + 4*A*b*c^2)*d^2*e^2 + (B*b^3 - 6*A*b^2*c)*d*e
^3 + (64*B*c^3*d^3*e - 24*(3*B*b*c^2 + 2*A*c^3)*d^2*e^2 + 12*(B*b^2*c + 4*A*b*c^2)*d*e^3 + (B*b^3 - 6*A*b^2*c)
*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (8*B*c^3*e^4*x^3 + 96*B*c^3*d^3*e - 12*(7*B*b*c^2
+ 6*A*c^3)*d^2*e^2 + 3*(B*b^2*c + 18*A*b*c^2)*d*e^3 - 2*(8*B*c^3*d*e^3 - (7*B*b*c^2 + 6*A*c^3)*e^4)*x^2 + (48
*B*c^3*d^2*e^2 - 2*(23*B*b*c^2 + 18*A*c^3)*d*e^3 + 3*(B*b^2*c + 10*A*b*c^2)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*e^
6*x + c^2*d*e^5)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.06, size = 4283, normalized size = 13.26 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e/(b*e-c*d)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c*A+1/4*B/e^2*((x+d/e)^2*c-(b*e-c*d)*d
/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b+2*B/e^5*d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*
(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c-6
/e^4/(b*e-c*d)*d^3*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e
)^(1/2))*c^(3/2)*b*B-3/2/e^3/(b*e-c*d)*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^3*B-3/2/e^
2/(b*e-c*d)*d*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2*A+6/e^3/(b*e-c*d)*d^2*ln(((x+d/e
)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(3/2)*b*A-1/e^2/(b
*e-c*d)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c*B*d+3/8/e/(b*e-c*d)*ln(((x+d/e)*c+1/2*(b*e
-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/c^(1/2)*b^3*A-3/e^4/(b*e-c*d)*d^
3*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(5/2)*
A+1/e*c/(b*e-c*d)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*x*B-c/(b*e-c*d)/d*((x+d/e)^2*c-(b*
e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*x*A+9/4/e/(b*e-c*d)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e)^(1/2)*b^2*A-1/e/(b*e-c*d)/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(5/2)*B-1/(b*e-c*d)/
d*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b*A+1/8*B/e^2/c*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-
2*c*d)*(x+d/e)/e)^(1/2)*b^2-3/2/e^2/(b*e-c*d)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b*c*
B*d-6/e^3/(b*e-c*d)*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/
e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*c*A+6/e^4/(b*e-c*d)*d^3/(-(
b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*
e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*c*B+15/2/e^4/(b*e-c*d)*d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln
((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)
*(x+d/e)/e)^(1/2))/(x+d/e))*b*c^2*A-15/2/e^5/(b*e-c*d)*d^4/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*
e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/
e))*b*c^2*B+1/3*B/e^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)-1/2*B/e^3*((x+d/e)^2*c-(b*e-c*
d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c*d-3/8*B/e^3*d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-
(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/c^(1/2)*b^2+3/2*B/e^4*d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/
2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b+3/e^5/(b*e-c*d)*d^4*ln(((x+d/e)*c+1/2*
(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(5/2)*B-9/4/e^2/(b*e-c*d)*
((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*B*d+3/e^3/(b*e-c*d)*d^2*((x+d/e)^2*c-(b*e-c*d)*d
/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^2*A-3/e^4/(b*e-c*d)*d^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e
)^(1/2)*c^2*B+3/2/e^3/(b*e-c*d)*d^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2*B+21/4/e^3
/(b*e-c*d)*d^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c*B-3/e^5/(b*e-c*d)*d^4/(-(b*e-c*d)
*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d
/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c^3*A+27/8/e^3/(b*e-c*d)*d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1
/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b^2*B+3/2/e/(b*e-c*d)*((x+d/e)^2*c-(b*e
-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b*c*A-3/8/e^2/(b*e-c*d)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((
x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/c^(1/2)*b^3*B*d-27/8/e^2/(b*e-c*d)*d*ln(((x+d/e)*c+1/
2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b^2*A-21/4/e^2/(b*
e-c*d)*d*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c*A+3/2/e^2/(b*e-c*d)*d/(-(b*e-c*d)*d/e^2
)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(
b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^3*A+3/e^6/(b*e-c*d)*d^5/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^
2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/
(x+d/e))*c^3*B-5/4*B/e^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*d-1/16*B/e^2/c^(3/2)*ln((
(x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b^3+B/e^4*d^2*
((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c-B/e^5*d^3*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)
+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(3/2)+1/e/(b*e-c*d)*((x+d/e)^2*c-(b*e-c*d)*d/e^2
+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b*B+1/(b*e-c*d)/d/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(5
/2)*A-B/e^4*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/
2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2-B/e^6*d^4/(-(b*e-c*d)*d/e^2)^(1/2)*
ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*
d)*(x+d/e)/e)^(1/2))/(x+d/e))*c^2

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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)^(3/2)/(e*x+d)^2,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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d zero or nonzero?

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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 )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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