3.18.76 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=345 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac {15 b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]

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Rubi [A]  time = 0.25, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac {15 b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(15*b^4*(b*d - a*e)^2*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - ((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^3) + (3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d +
e*x)^2) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) - (3*b^5*(b*d - a*e)
*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (b^6*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(3*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a + b*x))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 770

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

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^4} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^4} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {15 b^4 (b d-a e)^2}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^4}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^3}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^2}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)}-\frac {6 b^5 (b d-a e) (d+e x)}{e^6}+\frac {b^6 (d+e x)^2}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {15 b^4 (b d-a e)^2 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {3 b^5 (b d-a e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {b^6 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 320, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+3 a^5 b e^5 (d+3 e x)+15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^2 b^4 e^2 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )-3 a b^5 e \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )+60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (37 d^6+51 d^5 e x-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+3 d e^5 x^5-e^6 x^6\right )\right )}{3 e^7 (a+b x) (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

-1/3*(Sqrt[(a + b*x)^2]*(a^6*e^6 + 3*a^5*b*e^5*(d + 3*e*x) + 15*a^4*b^2*e^4*(d^2 + 3*d*e*x + 3*e^2*x^2) - 10*a
^3*b^3*d*e^3*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 15*a^2*b^4*e^2*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*
x^3 - 3*e^4*x^4) - 3*a*b^5*e*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5)
+ b^6*(37*d^6 + 51*d^5*e*x - 39*d^4*e^2*x^2 - 73*d^3*e^3*x^3 - 15*d^2*e^4*x^4 + 3*d*e^5*x^5 - e^6*x^6) + 60*b
^3*(b*d - a*e)^3*(d + e*x)^3*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^3)

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IntegrateAlgebraic [F]  time = 6.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

Defer[IntegrateAlgebraic][((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4, x]

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fricas [B]  time = 0.41, size = 576, normalized size = 1.67 \begin {gather*} \frac {b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \, {\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + {\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \, {\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \, {\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} + {\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/3*(b^6*e^6*x^6 - 37*b^6*d^6 + 141*a*b^5*d^5*e - 195*a^2*b^4*d^4*e^2 + 110*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e
^4 - 3*a^5*b*d*e^5 - a^6*e^6 - 3*(b^6*d*e^5 - 3*a*b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 3*a*b^5*d*e^5 + 3*a^2*b^4*e
^6)*x^4 + (73*b^6*d^3*e^3 - 189*a*b^5*d^2*e^4 + 135*a^2*b^4*d*e^5)*x^3 + 3*(13*b^6*d^4*e^2 - 9*a*b^5*d^3*e^3 -
45*a^2*b^4*d^2*e^4 + 60*a^3*b^3*d*e^5 - 15*a^4*b^2*e^6)*x^2 - 3*(17*b^6*d^5*e - 81*a*b^5*d^4*e^2 + 135*a^2*b^
4*d^3*e^3 - 90*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - 3*a*b^5*d^5*e + 3*a^2*b^4*d
^4*e^2 - a^3*b^3*d^3*e^3 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 3*(b^6*d^4*e^
2 - 3*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 - a^3*b^3*d*e^5)*x^2 + 3*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*
e^3 - a^3*b^3*d^2*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [A]  time = 0.19, size = 503, normalized size = 1.46 \begin {gather*} -20 \, {\left (b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{3} \, {\left (b^{6} x^{3} e^{8} \mathrm {sgn}\left (b x + a\right ) - 6 \, b^{6} d x^{2} e^{7} \mathrm {sgn}\left (b x + a\right ) + 30 \, b^{6} d^{2} x e^{6} \mathrm {sgn}\left (b x + a\right ) + 9 \, a b^{5} x^{2} e^{8} \mathrm {sgn}\left (b x + a\right ) - 72 \, a b^{5} d x e^{7} \mathrm {sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} x e^{8} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac {{\left (37 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 141 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 195 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 110 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 45 \, {\left (b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 9 \, {\left (9 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 35 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{3} b^{3} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-20*(b^6*d^3*sgn(b*x + a) - 3*a*b^5*d^2*e*sgn(b*x + a) + 3*a^2*b^4*d*e^2*sgn(b*x + a) - a^3*b^3*e^3*sgn(b*x +
a))*e^(-7)*log(abs(x*e + d)) + 1/3*(b^6*x^3*e^8*sgn(b*x + a) - 6*b^6*d*x^2*e^7*sgn(b*x + a) + 30*b^6*d^2*x*e^6
*sgn(b*x + a) + 9*a*b^5*x^2*e^8*sgn(b*x + a) - 72*a*b^5*d*x*e^7*sgn(b*x + a) + 45*a^2*b^4*x*e^8*sgn(b*x + a))*
e^(-12) - 1/3*(37*b^6*d^6*sgn(b*x + a) - 141*a*b^5*d^5*e*sgn(b*x + a) + 195*a^2*b^4*d^4*e^2*sgn(b*x + a) - 110
*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) + 3*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x
+ a) + 45*(b^6*d^4*e^2*sgn(b*x + a) - 4*a*b^5*d^3*e^3*sgn(b*x + a) + 6*a^2*b^4*d^2*e^4*sgn(b*x + a) - 4*a^3*b
^3*d*e^5*sgn(b*x + a) + a^4*b^2*e^6*sgn(b*x + a))*x^2 + 9*(9*b^6*d^5*e*sgn(b*x + a) - 35*a*b^5*d^4*e^2*sgn(b*x
+ a) + 50*a^2*b^4*d^3*e^3*sgn(b*x + a) - 30*a^3*b^3*d^2*e^4*sgn(b*x + a) + 5*a^4*b^2*d*e^5*sgn(b*x + a) + a^5
*b*e^6*sgn(b*x + a))*x)*e^(-7)/(x*e + d)^3

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maple [B]  time = 0.11, size = 692, normalized size = 2.01 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (b^{6} e^{6} x^{6}+9 a \,b^{5} e^{6} x^{5}-3 b^{6} d \,e^{5} x^{5}+60 a^{3} b^{3} e^{6} x^{3} \ln \left (e x +d \right )-180 a^{2} b^{4} d \,e^{5} x^{3} \ln \left (e x +d \right )+45 a^{2} b^{4} e^{6} x^{4}+180 a \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )-45 a \,b^{5} d \,e^{5} x^{4}-60 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )+15 b^{6} d^{2} e^{4} x^{4}+180 a^{3} b^{3} d \,e^{5} x^{2} \ln \left (e x +d \right )-540 a^{2} b^{4} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+135 a^{2} b^{4} d \,e^{5} x^{3}+540 a \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-189 a \,b^{5} d^{2} e^{4} x^{3}-180 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )+73 b^{6} d^{3} e^{3} x^{3}-45 a^{4} b^{2} e^{6} x^{2}+180 a^{3} b^{3} d^{2} e^{4} x \ln \left (e x +d \right )+180 a^{3} b^{3} d \,e^{5} x^{2}-540 a^{2} b^{4} d^{3} e^{3} x \ln \left (e x +d \right )-135 a^{2} b^{4} d^{2} e^{4} x^{2}+540 a \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-27 a \,b^{5} d^{3} e^{3} x^{2}-180 b^{6} d^{5} e x \ln \left (e x +d \right )+39 b^{6} d^{4} e^{2} x^{2}-9 a^{5} b \,e^{6} x -45 a^{4} b^{2} d \,e^{5} x +60 a^{3} b^{3} d^{3} e^{3} \ln \left (e x +d \right )+270 a^{3} b^{3} d^{2} e^{4} x -180 a^{2} b^{4} d^{4} e^{2} \ln \left (e x +d \right )-405 a^{2} b^{4} d^{3} e^{3} x +180 a \,b^{5} d^{5} e \ln \left (e x +d \right )+243 a \,b^{5} d^{4} e^{2} x -60 b^{6} d^{6} \ln \left (e x +d \right )-51 b^{6} d^{5} e x -a^{6} e^{6}-3 a^{5} b d \,e^{5}-15 a^{4} b^{2} d^{2} e^{4}+110 a^{3} b^{3} d^{3} e^{3}-195 a^{2} b^{4} d^{4} e^{2}+141 a \,b^{5} d^{5} e -37 b^{6} d^{6}\right )}{3 \left (b x +a \right )^{5} \left (e x +d \right )^{3} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x)

[Out]

1/3*((b*x+a)^2)^(5/2)*(b^6*e^6*x^6+180*ln(e*x+d)*x^3*a*b^5*d^2*e^4-180*ln(e*x+d)*x^3*a^2*b^4*d*e^5+180*a^3*b^3
*d*e^5*x^2*ln(e*x+d)-540*a^2*b^4*d^2*e^4*x^2*ln(e*x+d)+540*a*b^5*d^3*e^3*x^2*ln(e*x+d)+180*a^3*b^3*d^2*e^4*x*l
n(e*x+d)+540*a*b^5*d^4*e^2*x*ln(e*x+d)-540*a^2*b^4*d^3*e^3*x*ln(e*x+d)-60*b^6*d^6*ln(e*x+d)+180*a*b^5*d^5*e*ln
(e*x+d)-27*a*b^5*d^3*e^3*x^2-45*a^4*b^2*d*e^5*x+270*a^3*b^3*d^2*e^4*x-405*a^2*b^4*d^3*e^3*x+243*a*b^5*d^4*e^2*
x-3*a^5*b*d*e^5-37*b^6*d^6-a^6*e^6-15*a^4*b^2*d^2*e^4+110*a^3*b^3*d^3*e^3-195*a^2*b^4*d^4*e^2+141*a*b^5*d^5*e-
180*a^2*b^4*d^4*e^2*ln(e*x+d)-45*a*b^5*d*e^5*x^4+135*a^2*b^4*d*e^5*x^3+73*b^6*d^3*e^3*x^3+9*a*b^5*e^6*x^5-3*b^
6*d*e^5*x^5+45*a^2*b^4*e^6*x^4+15*b^6*d^2*e^4*x^4+60*a^3*b^3*d^3*e^3*ln(e*x+d)-45*a^4*b^2*e^6*x^2+39*b^6*d^4*e
^2*x^2-9*a^5*b*e^6*x-189*a*b^5*d^2*e^4*x^3+180*a^3*b^3*d*e^5*x^2-135*a^2*b^4*d^2*e^4*x^2-180*b^6*d^4*e^2*x^2*l
n(e*x+d)-51*b^6*d^5*e*x-180*b^6*d^5*e*x*ln(e*x+d)+60*ln(e*x+d)*x^3*a^3*b^3*e^6-60*ln(e*x+d)*x^3*b^6*d^3*e^3)/(
b*x+a)^5/e^7/(e*x+d)^3

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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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^4,x)

[Out]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**4,x)

[Out]

Integral((a + b*x)*((a + b*x)**2)**(5/2)/(d + e*x)**4, x)

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