3.18.79 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=356 \[ -\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{6 e^7 (a+b x) (d+e x)^6}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}+\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^2}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3} \]

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Rubi [A]  time = 0.21, antiderivative size = 356, 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 {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^2}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3}-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac {6 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{6 e^7 (a+b x) (d+e x)^6}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} \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)^7,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^7*(a + b*x)*(d + e*x)^6) + (6*b*(b*d - a*e)^5*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^5) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7
*(a + b*x)*(d + e*x)^4) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^3) -
(15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (6*b^5*(b*d - a*e)*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (b^6*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)^7} \, 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)^7} \, 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)^7} \, 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 d+a e)^6}{e^6 (d+e x)^7}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac {b^6}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}+\frac {6 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {b^6 \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 = 0.17, size = 258, normalized size = 0.72 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left ((b d-a e) \left (10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+a^2 b^3 e^2 \left (57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right )+a b^4 e \left (87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )+60 b^6 (d+e x)^6 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^6} \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)^7,x]

[Out]

(Sqrt[(a + b*x)^2]*((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x +
225*e^2*x^2) + a^2*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*
x + 975*d^2*e^2*x^2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2
*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) + 60*b^6*(d + e*x)^6*Log[d + e*x]))/(60*e^7*(a + b*x)*(d + e*x)^6)

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IntegrateAlgebraic [F]  time = 180.26, 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)^7,x]

[Out]

$Aborted

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fricas [A]  time = 0.42, size = 492, normalized size = 1.38 \begin {gather*} \frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + 6*b^6*d*e^5*x^5 +
15*b^6*d^2*e^4*x^4 + 20*b^6*d^3*e^3*x^3 + 15*b^6*d^4*e^2*x^2 + 6*b^6*d^5*e*x + b^6*d^6)*log(e*x + d))/(e^13*x
^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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giac [A]  time = 0.19, size = 507, normalized size = 1.42 \begin {gather*} b^{6} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (360 \, {\left (b^{6} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a b^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{2} b^{4} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{2} b^{4} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 2 \, a^{3} b^{3} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 12 \, a b^{5} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{2} b^{4} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, a^{4} b^{2} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 60 \, a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 12 \, a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right )\right )} x + {\left (147 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 60 \, 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 ) - 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 ) - 12 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

b^6*e^(-7)*log(abs(x*e + d))*sgn(b*x + a) + 1/60*(360*(b^6*d*e^4*sgn(b*x + a) - a*b^5*e^5*sgn(b*x + a))*x^5 +
450*(3*b^6*d^2*e^3*sgn(b*x + a) - 2*a*b^5*d*e^4*sgn(b*x + a) - a^2*b^4*e^5*sgn(b*x + a))*x^4 + 200*(11*b^6*d^3
*e^2*sgn(b*x + a) - 6*a*b^5*d^2*e^3*sgn(b*x + a) - 3*a^2*b^4*d*e^4*sgn(b*x + a) - 2*a^3*b^3*e^5*sgn(b*x + a))*
x^3 + 75*(25*b^6*d^4*e*sgn(b*x + a) - 12*a*b^5*d^3*e^2*sgn(b*x + a) - 6*a^2*b^4*d^2*e^3*sgn(b*x + a) - 4*a^3*b
^3*d*e^4*sgn(b*x + a) - 3*a^4*b^2*e^5*sgn(b*x + a))*x^2 + 6*(137*b^6*d^5*sgn(b*x + a) - 60*a*b^5*d^4*e*sgn(b*x
+ a) - 30*a^2*b^4*d^3*e^2*sgn(b*x + a) - 20*a^3*b^3*d^2*e^3*sgn(b*x + a) - 15*a^4*b^2*d*e^4*sgn(b*x + a) - 12
*a^5*b*e^5*sgn(b*x + a))*x + (147*b^6*d^6*sgn(b*x + a) - 60*a*b^5*d^5*e*sgn(b*x + a) - 30*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) - 12*a^5*b*d*e^5*sgn(b*x + a) - 1
0*a^6*e^6*sgn(b*x + a))*e^(-1))*e^(-6)/(x*e + d)^6

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maple [A]  time = 0.07, size = 507, normalized size = 1.42 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 b^{6} e^{6} x^{6} \ln \left (e x +d \right )+360 b^{6} d \,e^{5} x^{5} \ln \left (e x +d \right )-360 a \,b^{5} e^{6} x^{5}+900 b^{6} d^{2} e^{4} x^{4} \ln \left (e x +d \right )+360 b^{6} d \,e^{5} x^{5}-450 a^{2} b^{4} e^{6} x^{4}-900 a \,b^{5} d \,e^{5} x^{4}+1200 b^{6} d^{3} e^{3} x^{3} \ln \left (e x +d \right )+1350 b^{6} d^{2} e^{4} x^{4}-400 a^{3} b^{3} e^{6} x^{3}-600 a^{2} b^{4} d \,e^{5} x^{3}-1200 a \,b^{5} d^{2} e^{4} x^{3}+900 b^{6} d^{4} e^{2} x^{2} \ln \left (e x +d \right )+2200 b^{6} d^{3} e^{3} x^{3}-225 a^{4} b^{2} e^{6} x^{2}-300 a^{3} b^{3} d \,e^{5} x^{2}-450 a^{2} b^{4} d^{2} e^{4} x^{2}-900 a \,b^{5} d^{3} e^{3} x^{2}+360 b^{6} d^{5} e x \ln \left (e x +d \right )+1875 b^{6} d^{4} e^{2} x^{2}-72 a^{5} b \,e^{6} x -90 a^{4} b^{2} d \,e^{5} x -120 a^{3} b^{3} d^{2} e^{4} x -180 a^{2} b^{4} d^{3} e^{3} x -360 a \,b^{5} d^{4} e^{2} x +60 b^{6} d^{6} \ln \left (e x +d \right )+822 b^{6} d^{5} e x -10 a^{6} e^{6}-12 a^{5} b d \,e^{5}-15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}-30 a^{2} b^{4} d^{4} e^{2}-60 a \,b^{5} d^{5} e +147 b^{6} d^{6}\right )}{60 \left (b x +a \right )^{5} \left (e x +d \right )^{6} 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)^7,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(60*b^6*d^6*ln(e*x+d)+60*ln(e*x+d)*x^6*b^6*e^6-900*a*b^5*d^3*e^3*x^2-90*a^4*b^2*d*e^5*x
-120*a^3*b^3*d^2*e^4*x-180*a^2*b^4*d^3*e^3*x-360*a*b^5*d^4*e^2*x-12*a^5*b*d*e^5+147*b^6*d^6-10*a^6*e^6-15*a^4*
b^2*d^2*e^4-20*a^3*b^3*d^3*e^3-30*a^2*b^4*d^4*e^2-60*a*b^5*d^5*e-900*a*b^5*d*e^5*x^4-600*a^2*b^4*d*e^5*x^3+220
0*b^6*d^3*e^3*x^3-360*a*b^5*e^6*x^5+360*b^6*d*e^5*x^5-450*a^2*b^4*e^6*x^4+1350*b^6*d^2*e^4*x^4-400*a^3*b^3*e^6
*x^3-225*a^4*b^2*e^6*x^2+1875*b^6*d^4*e^2*x^2-72*a^5*b*e^6*x-1200*a*b^5*d^2*e^4*x^3-300*a^3*b^3*d*e^5*x^2-450*
a^2*b^4*d^2*e^4*x^2+360*b^6*d*e^5*x^5*ln(e*x+d)+900*b^6*d^4*e^2*x^2*ln(e*x+d)+900*b^6*d^2*e^4*x^4*ln(e*x+d)+82
2*b^6*d^5*e*x+360*b^6*d^5*e*x*ln(e*x+d)+1200*b^6*d^3*e^3*x^3*ln(e*x+d))/(b*x+a)^5/e^7/(e*x+d)^6

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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)^7,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 )}^7} \,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)^7,x)

[Out]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^7, x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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)**7,x)

[Out]

Exception raised: HeuristicGCDFailed

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