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

Optimal. Leaf size=266 \[ \frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac {10 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac {5 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac {e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]

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Rubi [A]  time = 0.30, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {646, 43} \begin {gather*} \frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{2 b^6}+\frac {10 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{9 b^6}+\frac {5 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{4 b^6}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)^5}{6 b^6}+\frac {e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)^5*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^6*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(7*b^6) + (5*e^2*(b*d - a*e)^3*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^6) + (10*e^
3*(b*d - a*e)^2*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^4*(b*d - a*e)*(a + b*x)^9*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*b^6) + (e^5*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*b^6)

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 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (d+e x)^5 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(b d-a e)^5 \left (a b+b^2 x\right )^5}{b^5}+\frac {5 e (b d-a e)^4 \left (a b+b^2 x\right )^6}{b^6}+\frac {10 e^2 (b d-a e)^3 \left (a b+b^2 x\right )^7}{b^7}+\frac {10 e^3 (b d-a e)^2 \left (a b+b^2 x\right )^8}{b^8}+\frac {5 e^4 (b d-a e) \left (a b+b^2 x\right )^9}{b^9}+\frac {e^5 \left (a b+b^2 x\right )^{10}}{b^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(b d-a e)^5 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac {5 e^2 (b d-a e)^3 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac {e^4 (b d-a e) (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^6}+\frac {e^5 (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 385, normalized size = 1.45 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (462 a^5 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+330 a^4 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+165 a^3 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+55 a^2 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+11 a b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )\right )}{2772 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(462*a^5*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) +
330*a^4*b*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + 165*a^3*b^2
*x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 55*a^2*b^3*x^3*
(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 11*a*b^4*x^4*(252*d
^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + b^5*x^5*(462*d^5 + 19
80*d^4*e*x + 3465*d^3*e^2*x^2 + 3080*d^2*e^3*x^3 + 1386*d*e^4*x^4 + 252*e^5*x^5)))/(2772*(a + b*x))

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IntegrateAlgebraic [F]  time = 4.16, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 427, normalized size = 1.61 \begin {gather*} \frac {1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac {1}{2} \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac {5}{4} \, {\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac {5}{7} \, {\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} + {\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b d^{5} + a^{5} d^{4} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^5*e^5*x^11 + a^5*d^5*x + 1/2*(b^5*d*e^4 + a*b^4*e^5)*x^10 + 5/9*(2*b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 2*a^2*
b^3*e^5)*x^9 + 5/4*(b^5*d^3*e^2 + 5*a*b^4*d^2*e^3 + 5*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^8 + 5/7*(b^5*d^4*e + 10*a
*b^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + a^4*b*e^5)*x^7 + 1/6*(b^5*d^5 + 25*a*b^4*d^4*e + 100*a^
2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + a^5*e^5)*x^6 + (a*b^4*d^5 + 10*a^2*b^3*d^4*e + 20*a^3*b
^2*d^3*e^2 + 10*a^4*b*d^2*e^3 + a^5*d*e^4)*x^5 + 5/2*(a^2*b^3*d^5 + 5*a^3*b^2*d^4*e + 5*a^4*b*d^3*e^2 + a^5*d^
2*e^3)*x^4 + 5/3*(2*a^3*b^2*d^5 + 5*a^4*b*d^4*e + 2*a^5*d^3*e^2)*x^3 + 5/2*(a^4*b*d^5 + a^5*d^4*e)*x^2

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giac [B]  time = 0.21, size = 686, normalized size = 2.58 \begin {gather*} \frac {1}{11} \, b^{5} x^{11} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{5} d x^{10} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, b^{5} d^{2} x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, b^{5} d^{3} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, b^{5} d^{4} x^{7} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a b^{4} x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{9} \, a b^{4} d x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{4} \, a b^{4} d^{2} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{7} \, a b^{4} d^{3} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{6} \, a b^{4} d^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + a b^{4} d^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, a^{2} b^{3} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{4} \, a^{2} b^{3} d x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {100}{7} \, a^{2} b^{3} d^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{3} \, a^{2} b^{3} d^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{4} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{4} \, a^{3} b^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{7} \, a^{3} b^{2} d x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {50}{3} \, a^{3} b^{2} d^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{2} d^{3} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{3} b^{2} d^{4} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{7} \, a^{4} b x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{6} \, a^{4} b d x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{4} b d^{2} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{4} b d^{3} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{3} \, a^{4} b d^{4} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{5} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{5} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{5} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{5} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{5} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{5} d^{5} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^5*x^11*e^5*sgn(b*x + a) + 1/2*b^5*d*x^10*e^4*sgn(b*x + a) + 10/9*b^5*d^2*x^9*e^3*sgn(b*x + a) + 5/4*b^5
*d^3*x^8*e^2*sgn(b*x + a) + 5/7*b^5*d^4*x^7*e*sgn(b*x + a) + 1/6*b^5*d^5*x^6*sgn(b*x + a) + 1/2*a*b^4*x^10*e^5
*sgn(b*x + a) + 25/9*a*b^4*d*x^9*e^4*sgn(b*x + a) + 25/4*a*b^4*d^2*x^8*e^3*sgn(b*x + a) + 50/7*a*b^4*d^3*x^7*e
^2*sgn(b*x + a) + 25/6*a*b^4*d^4*x^6*e*sgn(b*x + a) + a*b^4*d^5*x^5*sgn(b*x + a) + 10/9*a^2*b^3*x^9*e^5*sgn(b*
x + a) + 25/4*a^2*b^3*d*x^8*e^4*sgn(b*x + a) + 100/7*a^2*b^3*d^2*x^7*e^3*sgn(b*x + a) + 50/3*a^2*b^3*d^3*x^6*e
^2*sgn(b*x + a) + 10*a^2*b^3*d^4*x^5*e*sgn(b*x + a) + 5/2*a^2*b^3*d^5*x^4*sgn(b*x + a) + 5/4*a^3*b^2*x^8*e^5*s
gn(b*x + a) + 50/7*a^3*b^2*d*x^7*e^4*sgn(b*x + a) + 50/3*a^3*b^2*d^2*x^6*e^3*sgn(b*x + a) + 20*a^3*b^2*d^3*x^5
*e^2*sgn(b*x + a) + 25/2*a^3*b^2*d^4*x^4*e*sgn(b*x + a) + 10/3*a^3*b^2*d^5*x^3*sgn(b*x + a) + 5/7*a^4*b*x^7*e^
5*sgn(b*x + a) + 25/6*a^4*b*d*x^6*e^4*sgn(b*x + a) + 10*a^4*b*d^2*x^5*e^3*sgn(b*x + a) + 25/2*a^4*b*d^3*x^4*e^
2*sgn(b*x + a) + 25/3*a^4*b*d^4*x^3*e*sgn(b*x + a) + 5/2*a^4*b*d^5*x^2*sgn(b*x + a) + 1/6*a^5*x^6*e^5*sgn(b*x
+ a) + a^5*d*x^5*e^4*sgn(b*x + a) + 5/2*a^5*d^2*x^4*e^3*sgn(b*x + a) + 10/3*a^5*d^3*x^3*e^2*sgn(b*x + a) + 5/2
*a^5*d^4*x^2*e*sgn(b*x + a) + a^5*d^5*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 506, normalized size = 1.90 \begin {gather*} \frac {\left (252 b^{5} e^{5} x^{10}+1386 x^{9} a \,b^{4} e^{5}+1386 x^{9} b^{5} d \,e^{4}+3080 x^{8} a^{2} b^{3} e^{5}+7700 x^{8} a \,b^{4} d \,e^{4}+3080 x^{8} b^{5} d^{2} e^{3}+3465 x^{7} a^{3} b^{2} e^{5}+17325 x^{7} a^{2} b^{3} d \,e^{4}+17325 x^{7} a \,b^{4} d^{2} e^{3}+3465 x^{7} b^{5} d^{3} e^{2}+1980 x^{6} a^{4} b \,e^{5}+19800 x^{6} a^{3} b^{2} d \,e^{4}+39600 x^{6} a^{2} b^{3} d^{2} e^{3}+19800 x^{6} a \,b^{4} d^{3} e^{2}+1980 x^{6} b^{5} d^{4} e +462 x^{5} a^{5} e^{5}+11550 x^{5} a^{4} b d \,e^{4}+46200 x^{5} a^{3} b^{2} d^{2} e^{3}+46200 x^{5} a^{2} b^{3} d^{3} e^{2}+11550 x^{5} a \,b^{4} d^{4} e +462 x^{5} b^{5} d^{5}+2772 a^{5} d \,e^{4} x^{4}+27720 a^{4} b \,d^{2} e^{3} x^{4}+55440 a^{3} b^{2} d^{3} e^{2} x^{4}+27720 a^{2} b^{3} d^{4} e \,x^{4}+2772 a \,b^{4} d^{5} x^{4}+6930 x^{3} a^{5} d^{2} e^{3}+34650 x^{3} a^{4} b \,d^{3} e^{2}+34650 x^{3} a^{3} b^{2} d^{4} e +6930 x^{3} a^{2} b^{3} d^{5}+9240 x^{2} a^{5} d^{3} e^{2}+23100 x^{2} a^{4} b \,d^{4} e +9240 x^{2} a^{3} b^{2} d^{5}+6930 x \,a^{5} d^{4} e +6930 x \,a^{4} b \,d^{5}+2772 a^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{2772 \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2772*x*(252*b^5*e^5*x^10+1386*a*b^4*e^5*x^9+1386*b^5*d*e^4*x^9+3080*a^2*b^3*e^5*x^8+7700*a*b^4*d*e^4*x^8+308
0*b^5*d^2*e^3*x^8+3465*a^3*b^2*e^5*x^7+17325*a^2*b^3*d*e^4*x^7+17325*a*b^4*d^2*e^3*x^7+3465*b^5*d^3*e^2*x^7+19
80*a^4*b*e^5*x^6+19800*a^3*b^2*d*e^4*x^6+39600*a^2*b^3*d^2*e^3*x^6+19800*a*b^4*d^3*e^2*x^6+1980*b^5*d^4*e*x^6+
462*a^5*e^5*x^5+11550*a^4*b*d*e^4*x^5+46200*a^3*b^2*d^2*e^3*x^5+46200*a^2*b^3*d^3*e^2*x^5+11550*a*b^4*d^4*e*x^
5+462*b^5*d^5*x^5+2772*a^5*d*e^4*x^4+27720*a^4*b*d^2*e^3*x^4+55440*a^3*b^2*d^3*e^2*x^4+27720*a^2*b^3*d^4*e*x^4
+2772*a*b^4*d^5*x^4+6930*a^5*d^2*e^3*x^3+34650*a^4*b*d^3*e^2*x^3+34650*a^3*b^2*d^4*e*x^3+6930*a^2*b^3*d^5*x^3+
9240*a^5*d^3*e^2*x^2+23100*a^4*b*d^4*e*x^2+9240*a^3*b^2*d^5*x^2+6930*a^5*d^4*e*x+6930*a^4*b*d^5*x+2772*a^5*d^5
)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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maxima [B]  time = 1.20, size = 815, normalized size = 3.06 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{5} x^{4}}{11 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d e^{4} x^{3}}{2 \, b^{2}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{5} x^{3}}{22 \, b^{3}} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d^{5} x - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{4} e x}{6 \, b} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{3} e^{2} x}{3 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d^{2} e^{3} x}{3 \, b^{3}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} d e^{4} x}{6 \, b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{5} x}{6 \, b^{5}} + \frac {10 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{2} e^{3} x^{2}}{9 \, b^{2}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d e^{4} x^{2}}{18 \, b^{3}} + \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{5} x^{2}}{198 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{5}}{6 \, b} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{4} e}{6 \, b^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} d^{3} e^{2}}{3 \, b^{3}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} d^{2} e^{3}}{3 \, b^{4}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} d e^{4}}{6 \, b^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{6} e^{5}}{6 \, b^{6}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{3} e^{2} x}{4 \, b^{2}} - \frac {55 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d^{2} e^{3} x}{36 \, b^{3}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} d e^{4} x}{36 \, b^{4}} - \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} e^{5} x}{396 \, b^{5}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} d^{4} e}{7 \, b^{2}} - \frac {45 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a d^{3} e^{2}}{28 \, b^{3}} + \frac {415 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} d^{2} e^{3}}{252 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} d e^{4}}{252 \, b^{5}} + \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{4} e^{5}}{2772 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^5*x^4/b^2 + 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d*e^4*x^3/b^2 - 3/22*(b
^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^5*x^3/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d^5*x - 5/6*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*a*d^4*e*x/b + 5/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^3*e^2*x/b^2 - 5/3*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*a^3*d^2*e^3*x/b^3 + 5/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*d*e^4*x/b^4 - 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*a^5*e^5*x/b^5 + 10/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^2*e^3*x^2/b^2 - 13/18*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*a*d*e^4*x^2/b^3 + 31/198*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^5*x^2/b^4 + 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*a*d^5/b - 5/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^4*e/b^2 + 5/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*
a^3*d^3*e^2/b^3 - 5/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*d^2*e^3/b^4 + 5/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^
5*d*e^4/b^5 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6*e^5/b^6 + 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^3*e^2*x/
b^2 - 55/36*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d^2*e^3*x/b^3 + 29/36*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*d*e^4*
x/b^4 - 65/396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^5*x/b^5 + 5/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*d^4*e/b^2 -
45/28*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*d^3*e^2/b^3 + 415/252*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*d^2*e^3/b^4
- 209/252*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*d*e^4/b^5 + 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^5/b^
6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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