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

Optimal. Leaf size=345 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^7 (a+b x)}+\frac {15 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)} \]

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Rubi [A]  time = 0.30, 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)^5}{5 e^7 (a+b x)}-\frac {3 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)}+\frac {15 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 \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)^2,x]

[Out]

(15*b^2*(b*d - a*e)^4*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])/(e^7*(a + b*x)*(d + e*x)) - (10*b^3*(b*d - a*e)^3*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a
+ b*x)) + (5*b^4*(b*d - a*e)^2*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) - (3*b^5*(b*d - a*e)
*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)) + (b^6*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(5*e^7*(a + b*x)) - (6*b*(b*d - a*e)^5*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)^2} \, 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)^2} \, 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)^2} \, 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^2 (b d-a e)^4}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^2}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^2}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^3}{e^6}+\frac {b^6 (d+e x)^4}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {15 b^2 (b d-a e)^4 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}}{e^7 (a+b x) (d+e x)}-\frac {10 b^3 (b d-a e)^3 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {5 b^4 (b d-a e)^2 (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {3 b^5 (b d-a e) (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}+\frac {b^6 (d+e x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac {6 b (b d-a e)^5 \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 = 0.20, size = 320, normalized size = 0.93 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (-10 a^6 e^6+60 a^5 b d e^5+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+100 a^3 b^3 e^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+50 a^2 b^4 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b^5 e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-60 b (d+e x) (b d-a e)^5 \log (d+e x)+b^6 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )\right )}{10 e^7 (a+b x) (d+e x)} \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)^2,x]

[Out]

(Sqrt[(a + b*x)^2]*(60*a^5*b*d*e^5 - 10*a^6*e^6 + 150*a^4*b^2*e^4*(-d^2 + d*e*x + e^2*x^2) + 100*a^3*b^3*e^3*(
2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 50*a^2*b^4*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3
+ e^4*x^4) + 5*a*b^5*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^6
*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) - 60*b*(b*
d - a*e)^5*(d + e*x)*Log[d + e*x]))/(10*e^7*(a + b*x)*(d + e*x))

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IntegrateAlgebraic [F]  time = 3.49, 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)^2} \, 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)^2,x]

[Out]

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

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fricas [A]  time = 0.42, size = 496, normalized size = 1.44 \begin {gather*} \frac {2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \, {\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \, {\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \, {\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \, {\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \, {\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} - a^{5} b d e^{5} + {\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{8} x + d 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)^2,x, algorithm="fricas")

[Out]

1/10*(2*b^6*e^6*x^6 - 10*b^6*d^6 + 60*a*b^5*d^5*e - 150*a^2*b^4*d^4*e^2 + 200*a^3*b^3*d^3*e^3 - 150*a^4*b^2*d^
2*e^4 + 60*a^5*b*d*e^5 - 10*a^6*e^6 - 3*(b^6*d*e^5 - 5*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 5*a*b^5*d*e^5 + 10*a^
2*b^4*e^6)*x^4 - 10*(b^6*d^3*e^3 - 5*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 - 10*a^3*b^3*e^6)*x^3 + 30*(b^6*d^4*e^2
- 5*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 - 10*a^3*b^3*d*e^5 + 5*a^4*b^2*e^6)*x^2 + 10*(5*b^6*d^5*e - 24*a*b^5*d^
4*e^2 + 45*a^2*b^4*d^3*e^3 - 40*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5)*x - 60*(b^6*d^6 - 5*a*b^5*d^5*e + 10*a^2*b
^4*d^4*e^2 - 10*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 - a^5*b*d*e^5 + (b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*a^2*b^4*
d^3*e^3 - 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 - a^5*b*e^6)*x)*log(e*x + d))/(e^8*x + d*e^7)

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giac [A]  time = 0.20, size = 519, normalized size = 1.50 \begin {gather*} -6 \, {\left (b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{10} \, {\left (2 \, b^{6} x^{5} e^{8} \mathrm {sgn}\left (b x + a\right ) - 5 \, b^{6} d x^{4} e^{7} \mathrm {sgn}\left (b x + a\right ) + 10 \, b^{6} d^{2} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) - 20 \, b^{6} d^{3} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 50 \, b^{6} d^{4} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{5} x^{4} e^{8} \mathrm {sgn}\left (b x + a\right ) - 40 \, a b^{5} d x^{3} e^{7} \mathrm {sgn}\left (b x + a\right ) + 90 \, a b^{5} d^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) - 240 \, a b^{5} d^{3} x e^{5} \mathrm {sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} x^{3} e^{8} \mathrm {sgn}\left (b x + a\right ) - 150 \, a^{2} b^{4} d x^{2} e^{7} \mathrm {sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x e^{6} \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{3} b^{3} x^{2} e^{8} \mathrm {sgn}\left (b x + a\right ) - 400 \, a^{3} b^{3} d x e^{7} \mathrm {sgn}\left (b x + a\right ) + 150 \, a^{4} b^{2} x e^{8} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-10\right )} - \frac {{\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, 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 ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{x e + d} \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)^2,x, algorithm="giac")

[Out]

-6*(b^6*d^5*sgn(b*x + a) - 5*a*b^5*d^4*e*sgn(b*x + a) + 10*a^2*b^4*d^3*e^2*sgn(b*x + a) - 10*a^3*b^3*d^2*e^3*s
gn(b*x + a) + 5*a^4*b^2*d*e^4*sgn(b*x + a) - a^5*b*e^5*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/10*(2*b^6*x^
5*e^8*sgn(b*x + a) - 5*b^6*d*x^4*e^7*sgn(b*x + a) + 10*b^6*d^2*x^3*e^6*sgn(b*x + a) - 20*b^6*d^3*x^2*e^5*sgn(b
*x + a) + 50*b^6*d^4*x*e^4*sgn(b*x + a) + 15*a*b^5*x^4*e^8*sgn(b*x + a) - 40*a*b^5*d*x^3*e^7*sgn(b*x + a) + 90
*a*b^5*d^2*x^2*e^6*sgn(b*x + a) - 240*a*b^5*d^3*x*e^5*sgn(b*x + a) + 50*a^2*b^4*x^3*e^8*sgn(b*x + a) - 150*a^2
*b^4*d*x^2*e^7*sgn(b*x + a) + 450*a^2*b^4*d^2*x*e^6*sgn(b*x + a) + 100*a^3*b^3*x^2*e^8*sgn(b*x + a) - 400*a^3*
b^3*d*x*e^7*sgn(b*x + a) + 150*a^4*b^2*x*e^8*sgn(b*x + a))*e^(-10) - (b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn
(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*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
) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)

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maple [B]  time = 0.07, size = 601, normalized size = 1.74 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (2 b^{6} e^{6} x^{6}+15 a \,b^{5} e^{6} x^{5}-3 b^{6} d \,e^{5} x^{5}+50 a^{2} b^{4} e^{6} x^{4}-25 a \,b^{5} d \,e^{5} x^{4}+5 b^{6} d^{2} e^{4} x^{4}+100 a^{3} b^{3} e^{6} x^{3}-100 a^{2} b^{4} d \,e^{5} x^{3}+50 a \,b^{5} d^{2} e^{4} x^{3}-10 b^{6} d^{3} e^{3} x^{3}+60 a^{5} b \,e^{6} x \ln \left (e x +d \right )-300 a^{4} b^{2} d \,e^{5} x \ln \left (e x +d \right )+150 a^{4} b^{2} e^{6} x^{2}+600 a^{3} b^{3} d^{2} e^{4} x \ln \left (e x +d \right )-300 a^{3} b^{3} d \,e^{5} x^{2}-600 a^{2} b^{4} d^{3} e^{3} x \ln \left (e x +d \right )+300 a^{2} b^{4} d^{2} e^{4} x^{2}+300 a \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-150 a \,b^{5} d^{3} e^{3} x^{2}-60 b^{6} d^{5} e x \ln \left (e x +d \right )+30 b^{6} d^{4} e^{2} x^{2}+60 a^{5} b d \,e^{5} \ln \left (e x +d \right )-300 a^{4} b^{2} d^{2} e^{4} \ln \left (e x +d \right )+150 a^{4} b^{2} d \,e^{5} x +600 a^{3} b^{3} d^{3} e^{3} \ln \left (e x +d \right )-400 a^{3} b^{3} d^{2} e^{4} x -600 a^{2} b^{4} d^{4} e^{2} \ln \left (e x +d \right )+450 a^{2} b^{4} d^{3} e^{3} x +300 a \,b^{5} d^{5} e \ln \left (e x +d \right )-240 a \,b^{5} d^{4} e^{2} x -60 b^{6} d^{6} \ln \left (e x +d \right )+50 b^{6} d^{5} e x -10 a^{6} e^{6}+60 a^{5} b d \,e^{5}-150 a^{4} b^{2} d^{2} e^{4}+200 a^{3} b^{3} d^{3} e^{3}-150 a^{2} b^{4} d^{4} e^{2}+60 a \,b^{5} d^{5} e -10 b^{6} d^{6}\right )}{10 \left (b x +a \right )^{5} \left (e x +d \right ) 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)^2,x)

[Out]

1/10*((b*x+a)^2)^(5/2)*(2*b^6*e^6*x^6+600*ln(e*x+d)*x*a^3*b^3*d^2*e^4+300*ln(e*x+d)*x*a*b^5*d^4*e^2-300*ln(e*x
+d)*x*a^4*b^2*d*e^5-600*ln(e*x+d)*x*a^2*b^4*d^3*e^3-60*b^6*d^6*ln(e*x+d)+300*a*b^5*d^5*e*ln(e*x+d)-150*a*b^5*d
^3*e^3*x^2+150*a^4*b^2*d*e^5*x-400*a^3*b^3*d^2*e^4*x+450*a^2*b^4*d^3*e^3*x-240*a*b^5*d^4*e^2*x+60*d*e^5*a^5*b-
10*b^6*d^6-10*a^6*e^6-150*a^4*b^2*d^2*e^4+200*a^3*b^3*d^3*e^3-150*a^2*b^4*d^4*e^2+60*a*b^5*d^5*e-600*a^2*b^4*d
^4*e^2*ln(e*x+d)-25*a*b^5*d*e^5*x^4-100*a^2*b^4*d*e^5*x^3-10*b^6*d^3*e^3*x^3+15*a*b^5*e^6*x^5-3*b^6*d*e^5*x^5+
50*a^2*b^4*e^6*x^4+5*b^6*d^2*e^4*x^4+100*a^3*b^3*e^6*x^3+600*a^3*b^3*d^3*e^3*ln(e*x+d)+150*a^4*b^2*e^6*x^2+30*
b^6*d^4*e^2*x^2+60*a^5*b*d*e^5*ln(e*x+d)-300*a^4*b^2*d^2*e^4*ln(e*x+d)+50*a*b^5*d^2*e^4*x^3-300*a^3*b^3*d*e^5*
x^2+300*a^2*b^4*d^2*e^4*x^2+50*b^6*d^5*e*x+60*ln(e*x+d)*x*a^5*b*e^6-60*ln(e*x+d)*x*b^6*d^5*e)/(b*x+a)^5/e^7/(e
*x+d)

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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)^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(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 )}^2} \,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)^2,x)

[Out]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(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 (a + b x\right ) \left (\left (a + b x\right )^{2}\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((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**2,x)

[Out]

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

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