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

Optimal. Leaf size=344 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac {b^6 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{e^6 (a+b x)}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)} \]

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Rubi [A]  time = 0.24, antiderivative size = 344, 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^5 x \sqrt {a^2+2 a b x+b^2 x^2} (5 b d-6 a e)}{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}{e^7 (a+b x) (d+e x)}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{2 e^7 (a+b x) (d+e x)^2}+\frac {2 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)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{4 e^7 (a+b x) (d+e x)^4}+\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^7 (a+b x)}+\frac {b^6 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (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)^5,x]

[Out]

-((b^5*(5*b*d - 6*a*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^6*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^4) + (2*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)^3) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a +
b*x)*(d + e*x)) + (15*b^4*(b*d - a*e)^2*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)^5} \, 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)^5} \, 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)^5} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (-\frac {b^5 (5 b d-6 a e)}{e^6}+\frac {b^6 x}{e^5}+\frac {(-b d+a e)^6}{e^6 (d+e x)^5}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^4}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^3}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^2}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {b^5 (5 b d-6 a e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^6 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {2 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)^3}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {15 b^4 (b d-a e)^2 \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 = 318, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)+b^6 \left (-57 d^6-168 d^5 e x-132 d^4 e^2 x^2+32 d^3 e^3 x^3+68 d^2 e^4 x^4+12 d e^5 x^5-2 e^6 x^6\right )\right )}{4 e^7 (a+b x) (d+e x)^4} \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)^5,x]

[Out]

-1/4*(Sqrt[(a + b*x)^2]*(a^6*e^6 + 2*a^5*b*e^5*(d + 4*e*x) + 5*a^4*b^2*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + 20*a^
3*b^3*e^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a^2*b^4*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 +
48*e^3*x^3) + 2*a*b^5*e*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)
+ b^6*(-57*d^6 - 168*d^5*e*x - 132*d^4*e^2*x^2 + 32*d^3*e^3*x^3 + 68*d^2*e^4*x^4 + 12*d*e^5*x^5 - 2*e^6*x^6)
- 60*b^4*(b*d - a*e)^2*(d + e*x)^4*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^4)

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IntegrateAlgebraic [F]  time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \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)^5,x]

[Out]

$Aborted

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fricas [B]  time = 0.43, size = 571, normalized size = 1.66 \begin {gather*} \frac {2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \, {\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \, {\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \, {\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \, {\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \, {\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \, {\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \, {\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/4*(2*b^6*e^6*x^6 + 57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e
^4 - 2*a^5*b*d*e^5 - a^6*e^6 - 12*(b^6*d*e^5 - 2*a*b^5*e^6)*x^5 - 4*(17*b^6*d^2*e^4 - 24*a*b^5*d*e^5)*x^4 - 16
*(2*b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 - 15*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 6*(22*b^6*d^4*e^2 - 84*a*b^5*d^3*e
^3 + 90*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 4*(42*b^6*d^5*e - 124*a*b^5*d^4*e^2 + 110*a^
2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a^4*b^2*d*e^5 - 2*a^5*b*e^6)*x + 60*(b^6*d^6 - 2*a*b^5*d^5*e + a^2*b^4*
d^4*e^2 + (b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(b^6*d^3*e^3 - 2*a*b^5*d^2*e^4 + a^2*b^4*d*e^5)*
x^3 + 6*(b^6*d^4*e^2 - 2*a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4)*x^2 + 4*(b^6*d^5*e - 2*a*b^5*d^4*e^2 + a^2*b^4*d^3*e
^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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giac [A]  time = 0.23, size = 504, normalized size = 1.47 \begin {gather*} 15 \, {\left (b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (b^{6} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, b^{6} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 12 \, a b^{5} x e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-10\right )} + \frac {{\left (57 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 154 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 125 \, 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 ) - 5 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 80 \, {\left (b^{6} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{5} \mathrm {sgn}\left (b x + a\right ) - a^{3} b^{3} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 30 \, {\left (7 \, b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a b^{5} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, 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} + 4 \, {\left (47 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 130 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 110 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20 \, 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 ) - 2 \, a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{4 \, {\left (x e + d\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

15*(b^6*d^2*sgn(b*x + a) - 2*a*b^5*d*e*sgn(b*x + a) + a^2*b^4*e^2*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/2
*(b^6*x^2*e^5*sgn(b*x + a) - 10*b^6*d*x*e^4*sgn(b*x + a) + 12*a*b^5*x*e^5*sgn(b*x + a))*e^(-10) + 1/4*(57*b^6*
d^6*sgn(b*x + a) - 154*a*b^5*d^5*e*sgn(b*x + a) + 125*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) - 5*a^4*b^2*d^2*e^4*sgn(b*x + a) - 2*a^5*b*d*e^5*sgn(b*x + a) - a^6*e^6*sgn(b*x + a) + 80*(b^6*d^3*e^3*
sgn(b*x + a) - 3*a*b^5*d^2*e^4*sgn(b*x + a) + 3*a^2*b^4*d*e^5*sgn(b*x + a) - a^3*b^3*e^6*sgn(b*x + a))*x^3 + 3
0*(7*b^6*d^4*e^2*sgn(b*x + a) - 20*a*b^5*d^3*e^3*sgn(b*x + a) + 18*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 + 4*(47*b^6*d^5*e*sgn(b*x + a) - 130*a*b^5*d^4*e^2*sgn(b*x +
a) + 110*a^2*b^4*d^3*e^3*sgn(b*x + a) - 20*a^3*b^3*d^2*e^4*sgn(b*x + a) - 5*a^4*b^2*d*e^5*sgn(b*x + a) - 2*a^5
*b*e^6*sgn(b*x + a))*x)*e^(-7)/(x*e + d)^4

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maple [B]  time = 0.08, size = 670, normalized size = 1.95 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (2 b^{6} e^{6} x^{6}+60 a^{2} b^{4} e^{6} x^{4} \ln \left (e x +d \right )-120 a \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )+24 a \,b^{5} e^{6} x^{5}+60 b^{6} d^{2} e^{4} x^{4} \ln \left (e x +d \right )-12 b^{6} d \,e^{5} x^{5}+240 a^{2} b^{4} d \,e^{5} x^{3} \ln \left (e x +d \right )-480 a \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+96 a \,b^{5} d \,e^{5} x^{4}+240 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-68 b^{6} d^{2} e^{4} x^{4}-80 a^{3} b^{3} e^{6} x^{3}+360 a^{2} b^{4} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+240 a^{2} b^{4} d \,e^{5} x^{3}-720 a \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-96 a \,b^{5} d^{2} e^{4} x^{3}+360 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-32 b^{6} d^{3} e^{3} x^{3}-30 a^{4} b^{2} e^{6} x^{2}-120 a^{3} b^{3} d \,e^{5} x^{2}+240 a^{2} b^{4} d^{3} e^{3} x \ln \left (e x +d \right )+540 a^{2} b^{4} d^{2} e^{4} x^{2}-480 a \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-504 a \,b^{5} d^{3} e^{3} x^{2}+240 b^{6} d^{5} e x \ln \left (e x +d \right )+132 b^{6} d^{4} e^{2} x^{2}-8 a^{5} b \,e^{6} x -20 a^{4} b^{2} d \,e^{5} x -80 a^{3} b^{3} d^{2} e^{4} x +60 a^{2} b^{4} d^{4} e^{2} \ln \left (e x +d \right )+440 a^{2} b^{4} d^{3} e^{3} x -120 a \,b^{5} d^{5} e \ln \left (e x +d \right )-496 a \,b^{5} d^{4} e^{2} x +60 b^{6} d^{6} \ln \left (e x +d \right )+168 b^{6} d^{5} e x -a^{6} e^{6}-2 a^{5} b d \,e^{5}-5 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+125 a^{2} b^{4} d^{4} e^{2}-154 a \,b^{5} d^{5} e +57 b^{6} d^{6}\right )}{4 \left (b x +a \right )^{5} \left (e x +d \right )^{4} 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)^5,x)

[Out]

1/4*((b*x+a)^2)^(5/2)*(2*b^6*e^6*x^6-120*ln(e*x+d)*x^4*a*b^5*d*e^5-480*a*b^5*d^2*e^4*x^3*ln(e*x+d)+240*a^2*b^4
*d*e^5*x^3*ln(e*x+d)+360*a^2*b^4*d^2*e^4*x^2*ln(e*x+d)-720*a*b^5*d^3*e^3*x^2*ln(e*x+d)-480*a*b^5*d^4*e^2*x*ln(
e*x+d)+240*a^2*b^4*d^3*e^3*x*ln(e*x+d)+60*b^6*d^6*ln(e*x+d)-120*a*b^5*d^5*e*ln(e*x+d)-504*a*b^5*d^3*e^3*x^2-20
*a^4*b^2*d*e^5*x-80*a^3*b^3*d^2*e^4*x+440*a^2*b^4*d^3*e^3*x-496*a*b^5*d^4*e^2*x-2*a^5*b*d*e^5+57*b^6*d^6-a^6*e
^6-5*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+125*a^2*b^4*d^4*e^2-154*a*b^5*d^5*e+60*a^2*b^4*d^4*e^2*ln(e*x+d)+96*a*
b^5*d*e^5*x^4+240*a^2*b^4*d*e^5*x^3-32*b^6*d^3*e^3*x^3+24*a*b^5*e^6*x^5-12*b^6*d*e^5*x^5-68*b^6*d^2*e^4*x^4-80
*a^3*b^3*e^6*x^3-30*a^4*b^2*e^6*x^2+132*b^6*d^4*e^2*x^2-8*a^5*b*e^6*x-96*a*b^5*d^2*e^4*x^3-120*a^3*b^3*d*e^5*x
^2+540*a^2*b^4*d^2*e^4*x^2+360*b^6*d^4*e^2*x^2*ln(e*x+d)+60*ln(e*x+d)*x^4*a^2*b^4*e^6+60*ln(e*x+d)*x^4*b^6*d^2
*e^4+168*b^6*d^5*e*x+240*b^6*d^5*e*x*ln(e*x+d)+240*b^6*d^3*e^3*x^3*ln(e*x+d))/(b*x+a)^5/e^7/(e*x+d)^4

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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)^5,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 )}^5} \,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)^5,x)

[Out]

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

[Out]

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

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