3.4.7 \(\int \frac {(b x+c x^2)^{5/2}}{(d+e x)^2} \, dx\) [307]

Optimal. Leaf size=314 \[ -\frac {5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac {5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 e^6} \]

[Out]

-5/24*(-6*c*e*x-7*b*e+8*c*d)*(c*x^2+b*x)^(3/2)/e^3-(c*x^2+b*x)^(5/2)/e/(e*x+d)+5/64*(-b^4*e^4-16*b^3*c*d*e^3+1
44*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(3/2)/e^6-5/2*d^(3/2)*(
-b*e+c*d)^(3/2)*(-b*e+2*c*d)*arctanh(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))/e^6-
5/64*(64*c^3*d^3-112*b*c^2*d^2*e+48*b^2*c*d*e^2-b^3*e^3-2*c*e*(b^2*e^2-16*b*c*d*e+16*c^2*d^2)*x)*(c*x^2+b*x)^(
1/2)/c/e^5

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Rubi [A]
time = 0.27, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {746, 828, 857, 634, 212, 738} \begin {gather*} -\frac {5 \sqrt {b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{64 c e^5}+\frac {5 \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac {5 d^{3/2} (c d-b e)^{3/2} (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^6}-\frac {5 \left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3 - 2*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*x)*Sq
rt[b*x + c*x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e - 6*c*e*x)*(b*x + c*x^2)^(3/2))/(24*e^3) - (b*x + c*x^2)^(5/2)
/(e*(d + e*x)) + (5*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*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^6) - (5*d^(3/2)*(c*d - b*e)^(3/2)*(2*c*d - b*e)*ArcTanh[(b*d + (2
*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^6)

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 746

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

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 {\left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx &=-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e}\\ &=-\frac {5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac {5 \int \frac {\left (-b c d (8 c d-7 b e)-c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{d+e x} \, dx}{16 c e^3}\\ &=-\frac {5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \int \frac {\frac {1}{2} b c d \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )+\frac {1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{64 c^2 e^5}\\ &=-\frac {5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac {\left (5 d^2 (c d-b e)^2 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 e^6}+\frac {\left (5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c e^6}\\ &=-\frac {5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {\left (5 d^2 (c d-b e)^2 (2 c d-b e)\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 )}{e^6}+\frac {\left (5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\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^6}\\ &=-\frac {5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^5}-\frac {5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac {\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac {5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac {5 d^{3/2} (c d-b e)^{3/2} (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^6}\\ \end {align*}

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Mathematica [A]
time = 1.35, size = 361, normalized size = 1.15 \begin {gather*} \frac {(x (b+c x))^{5/2} \left (\frac {e \sqrt {x} \left (15 b^3 e^3 (d+e x)+2 b^2 c e^2 \left (-360 d^2-205 d e x+59 e^2 x^2\right )+8 b c^2 e \left (210 d^3+110 d^2 e x-35 d e^2 x^2+17 e^3 x^3\right )-16 c^3 \left (60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right )\right )}{c (b+c x)^2 (d+e x)}-\frac {960 d^{3/2} \sqrt {-c d+b e} \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(b+c x)^{5/2}}-\frac {15 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{c^{3/2} (b+c x)^{5/2}}\right )}{192 e^6 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(15*b^3*e^3*(d + e*x) + 2*b^2*c*e^2*(-360*d^2 - 205*d*e*x + 59*e^2*x^2) + 8*b
*c^2*e*(210*d^3 + 110*d^2*e*x - 35*d*e^2*x^2 + 17*e^3*x^3) - 16*c^3*(60*d^4 + 30*d^3*e*x - 10*d^2*e^2*x^2 + 5*
d*e^3*x^3 - 3*e^4*x^4)))/(c*(b + c*x)^2*(d + e*x)) - (960*d^(3/2)*Sqrt[-(c*d) + b*e]*(2*c^2*d^2 - 3*b*c*d*e +
b^2*e^2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(b + c*x)^(5/2
) - (15*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4)*Log[-(Sqrt[c]*Sqrt[x]
) + Sqrt[b + c*x]])/(c^(3/2)*(b + c*x)^(5/2))))/(192*e^6*x^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1454\) vs. \(2(282)=564\).
time = 0.52, size = 1455, normalized size = 4.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1455\)
risch	\(\text {Expression too large to display}\)	\(1906\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^2*(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))))))

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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((c*x^2+b*x)^(5/2)/(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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 3.42, size = 1815, normalized size = 5.78 \begin {gather*} \left [-\frac {15 \, {\left (128 \, c^{4} d^{5} - b^{4} x e^{5} - {\left (16 \, b^{3} c d x + b^{4} d\right )} e^{4} + 16 \, {\left (9 \, b^{2} c^{2} d^{2} x - b^{3} c d^{2}\right )} e^{3} - 16 \, {\left (16 \, b c^{3} d^{3} x - 9 \, b^{2} c^{2} d^{3}\right )} e^{2} + 128 \, {\left (c^{4} d^{4} x - 2 \, b c^{3} d^{4}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 960 \, {\left (2 \, c^{4} d^{4} + b^{2} c^{2} d x e^{3} - {\left (3 \, b c^{3} d^{2} x - b^{2} c^{2} d^{2}\right )} e^{2} + {\left (2 \, c^{4} d^{3} x - 3 \, b c^{3} d^{3}\right )} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {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}\right ) + 2 \, {\left (960 \, c^{4} d^{4} e - {\left (48 \, c^{4} x^{4} + 136 \, b c^{3} x^{3} + 118 \, b^{2} c^{2} x^{2} + 15 \, b^{3} c x\right )} e^{5} + 5 \, {\left (16 \, c^{4} d x^{3} + 56 \, b c^{3} d x^{2} + 82 \, b^{2} c^{2} d x - 3 \, b^{3} c d\right )} e^{4} - 80 \, {\left (2 \, c^{4} d^{2} x^{2} + 11 \, b c^{3} d^{2} x - 9 \, b^{2} c^{2} d^{2}\right )} e^{3} + 240 \, {\left (2 \, c^{4} d^{3} x - 7 \, b c^{3} d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{384 \, {\left (c^{2} x e^{7} + c^{2} d e^{6}\right )}}, -\frac {1920 \, {\left (2 \, c^{4} d^{4} + b^{2} c^{2} d x e^{3} - {\left (3 \, b c^{3} d^{2} x - b^{2} c^{2} d^{2}\right )} e^{2} + {\left (2 \, c^{4} d^{3} x - 3 \, b c^{3} d^{3}\right )} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) + 15 \, {\left (128 \, c^{4} d^{5} - b^{4} x e^{5} - {\left (16 \, b^{3} c d x + b^{4} d\right )} e^{4} + 16 \, {\left (9 \, b^{2} c^{2} d^{2} x - b^{3} c d^{2}\right )} e^{3} - 16 \, {\left (16 \, b c^{3} d^{3} x - 9 \, b^{2} c^{2} d^{3}\right )} e^{2} + 128 \, {\left (c^{4} d^{4} x - 2 \, b c^{3} d^{4}\right )} e\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (960 \, c^{4} d^{4} e - {\left (48 \, c^{4} x^{4} + 136 \, b c^{3} x^{3} + 118 \, b^{2} c^{2} x^{2} + 15 \, b^{3} c x\right )} e^{5} + 5 \, {\left (16 \, c^{4} d x^{3} + 56 \, b c^{3} d x^{2} + 82 \, b^{2} c^{2} d x - 3 \, b^{3} c d\right )} e^{4} - 80 \, {\left (2 \, c^{4} d^{2} x^{2} + 11 \, b c^{3} d^{2} x - 9 \, b^{2} c^{2} d^{2}\right )} e^{3} + 240 \, {\left (2 \, c^{4} d^{3} x - 7 \, b c^{3} d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{384 \, {\left (c^{2} x e^{7} + c^{2} d e^{6}\right )}}, -\frac {15 \, {\left (128 \, c^{4} d^{5} - b^{4} x e^{5} - {\left (16 \, b^{3} c d x + b^{4} d\right )} e^{4} + 16 \, {\left (9 \, b^{2} c^{2} d^{2} x - b^{3} c d^{2}\right )} e^{3} - 16 \, {\left (16 \, b c^{3} d^{3} x - 9 \, b^{2} c^{2} d^{3}\right )} e^{2} + 128 \, {\left (c^{4} d^{4} x - 2 \, b c^{3} d^{4}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - 480 \, {\left (2 \, c^{4} d^{4} + b^{2} c^{2} d x e^{3} - {\left (3 \, b c^{3} d^{2} x - b^{2} c^{2} d^{2}\right )} e^{2} + {\left (2 \, c^{4} d^{3} x - 3 \, b c^{3} d^{3}\right )} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {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}\right ) + {\left (960 \, c^{4} d^{4} e - {\left (48 \, c^{4} x^{4} + 136 \, b c^{3} x^{3} + 118 \, b^{2} c^{2} x^{2} + 15 \, b^{3} c x\right )} e^{5} + 5 \, {\left (16 \, c^{4} d x^{3} + 56 \, b c^{3} d x^{2} + 82 \, b^{2} c^{2} d x - 3 \, b^{3} c d\right )} e^{4} - 80 \, {\left (2 \, c^{4} d^{2} x^{2} + 11 \, b c^{3} d^{2} x - 9 \, b^{2} c^{2} d^{2}\right )} e^{3} + 240 \, {\left (2 \, c^{4} d^{3} x - 7 \, b c^{3} d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{192 \, {\left (c^{2} x e^{7} + c^{2} d e^{6}\right )}}, -\frac {960 \, {\left (2 \, c^{4} d^{4} + b^{2} c^{2} d x e^{3} - {\left (3 \, b c^{3} d^{2} x - b^{2} c^{2} d^{2}\right )} e^{2} + {\left (2 \, c^{4} d^{3} x - 3 \, b c^{3} d^{3}\right )} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) + 15 \, {\left (128 \, c^{4} d^{5} - b^{4} x e^{5} - {\left (16 \, b^{3} c d x + b^{4} d\right )} e^{4} + 16 \, {\left (9 \, b^{2} c^{2} d^{2} x - b^{3} c d^{2}\right )} e^{3} - 16 \, {\left (16 \, b c^{3} d^{3} x - 9 \, b^{2} c^{2} d^{3}\right )} e^{2} + 128 \, {\left (c^{4} d^{4} x - 2 \, b c^{3} d^{4}\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (960 \, c^{4} d^{4} e - {\left (48 \, c^{4} x^{4} + 136 \, b c^{3} x^{3} + 118 \, b^{2} c^{2} x^{2} + 15 \, b^{3} c x\right )} e^{5} + 5 \, {\left (16 \, c^{4} d x^{3} + 56 \, b c^{3} d x^{2} + 82 \, b^{2} c^{2} d x - 3 \, b^{3} c d\right )} e^{4} - 80 \, {\left (2 \, c^{4} d^{2} x^{2} + 11 \, b c^{3} d^{2} x - 9 \, b^{2} c^{2} d^{2}\right )} e^{3} + 240 \, {\left (2 \, c^{4} d^{3} x - 7 \, b c^{3} d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{192 \, {\left (c^{2} x e^{7} + c^{2} d e^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(15*(128*c^4*d^5 - b^4*x*e^5 - (16*b^3*c*d*x + b^4*d)*e^4 + 16*(9*b^2*c^2*d^2*x - b^3*c*d^2)*e^3 - 16*
(16*b*c^3*d^3*x - 9*b^2*c^2*d^3)*e^2 + 128*(c^4*d^4*x - 2*b*c^3*d^4)*e)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 +
b*x)*sqrt(c)) - 960*(2*c^4*d^4 + b^2*c^2*d*x*e^3 - (3*b*c^3*d^2*x - b^2*c^2*d^2)*e^2 + (2*c^4*d^3*x - 3*b*c^3
*d^3)*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*(960*c^4*d^4*e - (48*c^4*x^4 + 136*b*c^3*x^3 + 118*b^2*c^2*x^2 + 15*b^3*c*x)*e^5 + 5*(16*c^4*d*x^3 + 56*b*
c^3*d*x^2 + 82*b^2*c^2*d*x - 3*b^3*c*d)*e^4 - 80*(2*c^4*d^2*x^2 + 11*b*c^3*d^2*x - 9*b^2*c^2*d^2)*e^3 + 240*(2
*c^4*d^3*x - 7*b*c^3*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c^2*x*e^7 + c^2*d*e^6), -1/384*(1920*(2*c^4*d^4 + b^2*c^2*d
*x*e^3 - (3*b*c^3*d^2*x - b^2*c^2*d^2)*e^2 + (2*c^4*d^3*x - 3*b*c^3*d^3)*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(
-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e)) + 15*(128*c^4*d^5 - b^4*x*e^5 - (16*b^3*c*d*x + b^4*d)*e^4
+ 16*(9*b^2*c^2*d^2*x - b^3*c*d^2)*e^3 - 16*(16*b*c^3*d^3*x - 9*b^2*c^2*d^3)*e^2 + 128*(c^4*d^4*x - 2*b*c^3*d^
4)*e)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(960*c^4*d^4*e - (48*c^4*x^4 + 136*b*c^3*x^3 +
118*b^2*c^2*x^2 + 15*b^3*c*x)*e^5 + 5*(16*c^4*d*x^3 + 56*b*c^3*d*x^2 + 82*b^2*c^2*d*x - 3*b^3*c*d)*e^4 - 80*(2
*c^4*d^2*x^2 + 11*b*c^3*d^2*x - 9*b^2*c^2*d^2)*e^3 + 240*(2*c^4*d^3*x - 7*b*c^3*d^3)*e^2)*sqrt(c*x^2 + b*x))/(
c^2*x*e^7 + c^2*d*e^6), -1/192*(15*(128*c^4*d^5 - b^4*x*e^5 - (16*b^3*c*d*x + b^4*d)*e^4 + 16*(9*b^2*c^2*d^2*x
- b^3*c*d^2)*e^3 - 16*(16*b*c^3*d^3*x - 9*b^2*c^2*d^3)*e^2 + 128*(c^4*d^4*x - 2*b*c^3*d^4)*e)*sqrt(-c)*arctan
(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 480*(2*c^4*d^4 + b^2*c^2*d*x*e^3 - (3*b*c^3*d^2*x - b^2*c^2*d^2)*e^2 + (2
*c^4*d^3*x - 3*b*c^3*d^3)*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)) + (960*c^4*d^4*e - (48*c^4*x^4 + 136*b*c^3*x^3 + 118*b^2*c^2*x^2 + 15*b^3*c*x)*e^5 + 5*(16
*c^4*d*x^3 + 56*b*c^3*d*x^2 + 82*b^2*c^2*d*x - 3*b^3*c*d)*e^4 - 80*(2*c^4*d^2*x^2 + 11*b*c^3*d^2*x - 9*b^2*c^2
*d^2)*e^3 + 240*(2*c^4*d^3*x - 7*b*c^3*d^3)*e^2)*sqrt(c*x^2 + b*x))/(c^2*x*e^7 + c^2*d*e^6), -1/192*(960*(2*c^
4*d^4 + b^2*c^2*d*x*e^3 - (3*b*c^3*d^2*x - b^2*c^2*d^2)*e^2 + (2*c^4*d^3*x - 3*b*c^3*d^3)*e)*sqrt(-c*d^2 + b*d
*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e)) + 15*(128*c^4*d^5 - b^4*x*e^5 - (16*b^3*c*
d*x + b^4*d)*e^4 + 16*(9*b^2*c^2*d^2*x - b^3*c*d^2)*e^3 - 16*(16*b*c^3*d^3*x - 9*b^2*c^2*d^3)*e^2 + 128*(c^4*d
^4*x - 2*b*c^3*d^4)*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (960*c^4*d^4*e - (48*c^4*x^4 + 136*
b*c^3*x^3 + 118*b^2*c^2*x^2 + 15*b^3*c*x)*e^5 + 5*(16*c^4*d*x^3 + 56*b*c^3*d*x^2 + 82*b^2*c^2*d*x - 3*b^3*c*d)
*e^4 - 80*(2*c^4*d^2*x^2 + 11*b*c^3*d^2*x - 9*b^2*c^2*d^2)*e^3 + 240*(2*c^4*d^3*x - 7*b*c^3*d^3)*e^2)*sqrt(c*x
^2 + b*x))/(c^2*x*e^7 + c^2*d*e^6)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
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((c*x^2+b*x)^(5/2)/(e*x+d)^2,x, algorithm="giac")

[Out]

Timed out

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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 (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)^(5/2)/(d + e*x)^2,x)

[Out]

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

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