3.18.78 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=344 \[ -\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^3}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^2} \]

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Rubi [A]  time = 0.23, 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 {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) (d+e x)}+\frac {10 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)^2}-\frac {5 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)^3}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^4}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^5}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^7 (a+b x)}+\frac {b^6 x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (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)^6,x]

[Out]

(b^6*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])/(5*e^7*(
a + b*x)*(d + e*x)^5) + (3*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^4) - (5*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) + (10*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)^2) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(
a + b*x)*(d + e*x)) - (6*b^5*(b*d - a*e)*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)^6} \, 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)^6} \, 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)^6} \, 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^6}{e^6}+\frac {(-b d+a e)^6}{e^6 (d+e x)^6}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^5}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^4}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^3}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^2}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac {b^6 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}}{5 e^7 (a+b x) (d+e x)^5}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^4}-\frac {5 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}+\frac {10 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)^2}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {6 b^5 (b d-a e) \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 = 0.16, size = 315, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+30 a^2 b^4 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-a b^5 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+b^6 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )}{10 e^7 (a+b x) (d+e x)^5} \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)^6,x]

[Out]

-1/10*(Sqrt[(a + b*x)^2]*(2*a^6*e^6 + 3*a^5*b*e^5*(d + 5*e*x) + 5*a^4*b^2*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 1
0*a^3*b^3*e^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 30*a^2*b^4*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2
+ 10*d*e^3*x^3 + 5*e^4*x^4) - a*b^5*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x
^4) + b^6*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6*x
^6) + 60*b^5*(b*d - a*e)*(d + e*x)^5*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^5)

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IntegrateAlgebraic [F]  time = 180.12, 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)^6,x]

[Out]

$Aborted

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fricas [B]  time = 0.42, size = 542, normalized size = 1.58 \begin {gather*} \frac {10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, 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} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \, {\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \, {\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \, {\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \, {\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, 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} + 3 \, a^{5} b e^{6}\right )} x - 60 \, {\left (b^{6} d^{6} - a b^{5} d^{5} e + {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \, {\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/10*(10*b^6*e^6*x^6 + 50*b^6*d*e^5*x^5 - 87*b^6*d^6 + 137*a*b^5*d^5*e - 30*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 - 3*a^5*b*d*e^5 - 2*a^6*e^6 - 50*(b^6*d^2*e^4 - 6*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 - 10
0*(4*b^6*d^3*e^3 - 9*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 - 50*(12*b^6*d^4*e^2 - 22*a*b^5*d^3*e^
3 + 6*a^2*b^4*d^2*e^4 + 2*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 - 5*(75*b^6*d^5*e - 125*a*b^5*d^4*e^2 + 30*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 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - a*b^5*d^5*e + (b^6*d*e^5 - a*b
^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - a*b^5*d*e^5)*x^4 + 10*(b^6*d^3*e^3 - a*b^5*d^2*e^4)*x^3 + 10*(b^6*d^4*e^2 - a*b
^5*d^3*e^3)*x^2 + 5*(b^6*d^5*e - a*b^5*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 +
10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

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giac [A]  time = 0.20, size = 499, normalized size = 1.45 \begin {gather*} b^{6} x e^{\left (-6\right )} \mathrm {sgn}\left (b x + a\right ) - 6 \, {\left (b^{6} d \mathrm {sgn}\left (b x + a\right ) - a b^{5} e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (87 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 137 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, 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 ) + 3 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 150 \, {\left (b^{6} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 100 \, {\left (5 \, b^{6} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 9 \, 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} + 50 \, {\left (13 \, b^{6} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 22 \, 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 ) + 2 \, 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} + 5 \, {\left (77 \, b^{6} d^{5} e \mathrm {sgn}\left (b x + a\right ) - 125 \, a b^{5} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{2} b^{4} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, 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 ) + 3 \, a^{5} b e^{6} \mathrm {sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{10 \, {\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

b^6*x*e^(-6)*sgn(b*x + a) - 6*(b^6*d*sgn(b*x + a) - a*b^5*e*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) - 1/10*(87*
b^6*d^6*sgn(b*x + a) - 137*a*b^5*d^5*e*sgn(b*x + a) + 30*a^2*b^4*d^4*e^2*sgn(b*x + a) + 10*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) + 3*a^5*b*d*e^5*sgn(b*x + a) + 2*a^6*e^6*sgn(b*x + a) + 150*(b^6*d^
2*e^4*sgn(b*x + a) - 2*a*b^5*d*e^5*sgn(b*x + a) + a^2*b^4*e^6*sgn(b*x + a))*x^4 + 100*(5*b^6*d^3*e^3*sgn(b*x +
a) - 9*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 + 50*(13*b^6
*d^4*e^2*sgn(b*x + a) - 22*a*b^5*d^3*e^3*sgn(b*x + a) + 6*a^2*b^4*d^2*e^4*sgn(b*x + a) + 2*a^3*b^3*d*e^5*sgn(b
*x + a) + a^4*b^2*e^6*sgn(b*x + a))*x^2 + 5*(77*b^6*d^5*e*sgn(b*x + a) - 125*a*b^5*d^4*e^2*sgn(b*x + a) + 30*a
^2*b^4*d^3*e^3*sgn(b*x + a) + 10*a^3*b^3*d^2*e^4*sgn(b*x + a) + 5*a^4*b^2*d*e^5*sgn(b*x + a) + 3*a^5*b*e^6*sgn
(b*x + a))*x)*e^(-7)/(x*e + d)^5

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maple [B]  time = 0.11, size = 603, normalized size = 1.75 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 a \,b^{5} e^{6} x^{5} \ln \left (e x +d \right )-60 b^{6} d \,e^{5} x^{5} \ln \left (e x +d \right )+10 b^{6} e^{6} x^{6}+300 a \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )-300 b^{6} d^{2} e^{4} x^{4} \ln \left (e x +d \right )+50 b^{6} d \,e^{5} x^{5}-150 a^{2} b^{4} e^{6} x^{4}+600 a \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+300 a \,b^{5} d \,e^{5} x^{4}-600 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-50 b^{6} d^{2} e^{4} x^{4}-100 a^{3} b^{3} e^{6} x^{3}-300 a^{2} b^{4} d \,e^{5} x^{3}+600 a \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )+900 a \,b^{5} d^{2} e^{4} x^{3}-600 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-400 b^{6} d^{3} e^{3} x^{3}-50 a^{4} b^{2} e^{6} x^{2}-100 a^{3} b^{3} d \,e^{5} x^{2}-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 )+1100 a \,b^{5} d^{3} e^{3} x^{2}-300 b^{6} d^{5} e x \ln \left (e x +d \right )-600 b^{6} d^{4} e^{2} x^{2}-15 a^{5} b \,e^{6} x -25 a^{4} b^{2} d \,e^{5} x -50 a^{3} b^{3} d^{2} e^{4} x -150 a^{2} b^{4} d^{3} e^{3} x +60 a \,b^{5} d^{5} e \ln \left (e x +d \right )+625 a \,b^{5} d^{4} e^{2} x -60 b^{6} d^{6} \ln \left (e x +d \right )-375 b^{6} d^{5} e x -2 a^{6} e^{6}-3 a^{5} b d \,e^{5}-5 a^{4} b^{2} d^{2} e^{4}-10 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}+137 a \,b^{5} d^{5} e -87 b^{6} d^{6}\right )}{10 \left (b x +a \right )^{5} \left (e x +d \right )^{5} 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)^6,x)

[Out]

1/10*((b*x+a)^2)^(5/2)*(10*b^6*e^6*x^6+300*a*b^5*d*e^5*x^4*ln(e*x+d)+600*a*b^5*d^2*e^4*x^3*ln(e*x+d)+600*a*b^5
*d^3*e^3*x^2*ln(e*x+d)+300*a*b^5*d^4*e^2*x*ln(e*x+d)-60*b^6*d^6*ln(e*x+d)+60*a*b^5*d^5*e*ln(e*x+d)+1100*a*b^5*
d^3*e^3*x^2-25*a^4*b^2*d*e^5*x-50*a^3*b^3*d^2*e^4*x-150*a^2*b^4*d^3*e^3*x+625*a*b^5*d^4*e^2*x-3*a^5*b*d*e^5-87
*b^6*d^6-2*a^6*e^6-5*a^4*b^2*d^2*e^4-10*a^3*b^3*d^3*e^3-30*a^2*b^4*d^4*e^2+137*a*b^5*d^5*e+300*a*b^5*d*e^5*x^4
-300*a^2*b^4*d*e^5*x^3-400*b^6*d^3*e^3*x^3+50*b^6*d*e^5*x^5-150*a^2*b^4*e^6*x^4-50*b^6*d^2*e^4*x^4-100*a^3*b^3
*e^6*x^3+60*ln(e*x+d)*x^5*a*b^5*e^6-50*a^4*b^2*e^6*x^2-600*b^6*d^4*e^2*x^2-15*a^5*b*e^6*x+900*a*b^5*d^2*e^4*x^
3-100*a^3*b^3*d*e^5*x^2-300*a^2*b^4*d^2*e^4*x^2-60*ln(e*x+d)*x^5*b^6*d*e^5-600*b^6*d^4*e^2*x^2*ln(e*x+d)-300*b
^6*d^2*e^4*x^4*ln(e*x+d)-375*b^6*d^5*e*x-300*b^6*d^5*e*x*ln(e*x+d)-600*b^6*d^3*e^3*x^3*ln(e*x+d))/(b*x+a)^5/e^
7/(e*x+d)^5

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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)^6,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 )}^6} \,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)^6,x)

[Out]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^6, 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 )^{6}}\, 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)**6,x)

[Out]

Integral((a + b*x)*((a + b*x)**2)**(5/2)/(d + e*x)**6, x)

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