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

Optimal. Leaf size=263 \[ \frac {5 e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^2}{b^6}+\frac {10 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac {e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \]

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Rubi [A]  time = 0.38, antiderivative size = 263, 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 {5 e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac {e^3 \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^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac {e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(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)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^7*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(8*b^6) + (10*e^2*(b*d - a*e)^3*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^3*
(b*d - a*e)^2*(a + b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^6 + (5*e^4*(b*d - a*e)*(a + b*x)^10*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(11*b^6) + (e^5*(a + b*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*b^6)

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 (a+b x) (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 (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 (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 d-a e)^5 (a+b x)^6}{b^5}+\frac {5 e (b d-a e)^4 (a+b x)^7}{b^5}+\frac {10 e^2 (b d-a e)^3 (a+b x)^8}{b^5}+\frac {10 e^3 (b d-a e)^2 (a+b x)^9}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^{10}}{b^5}+\frac {e^5 (a+b x)^{11}}{b^5}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^5 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^6}+\frac {10 e^2 (b d-a e)^3 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac {e^3 (b d-a e)^2 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^6}+\frac {e^5 (a+b x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{12 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 448, normalized size = 1.70 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (924 a^6 \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 )+792 a^5 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 )+495 a^4 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 )+220 a^3 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 )+66 a^2 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 )+12 a 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 )+b^6 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )\right )}{5544 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(924*a^6*(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) +
792*a^5*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) + 495*a^4*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) + 220*a^3*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) + 66*a^2*b^4*x^4*(25
2*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) + 12*a*b^5*x^5*(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) + b^6*x^6*(792*d^5 +
3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5)))/(5544*(a + b*x))

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IntegrateAlgebraic [F]  time = 4.48, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (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[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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fricas [B]  time = 0.43, size = 517, normalized size = 1.97 \begin {gather*} \frac {1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac {1}{11} \, {\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^6*e^5*x^12 + a^6*d^5*x + 1/11*(5*b^6*d*e^4 + 6*a*b^5*e^5)*x^11 + 1/2*(2*b^6*d^2*e^3 + 6*a*b^5*d*e^4 + 3
*a^2*b^4*e^5)*x^10 + 5/9*(2*b^6*d^3*e^2 + 12*a*b^5*d^2*e^3 + 15*a^2*b^4*d*e^4 + 4*a^3*b^3*e^5)*x^9 + 5/8*(b^6*
d^4*e + 12*a*b^5*d^3*e^2 + 30*a^2*b^4*d^2*e^3 + 20*a^3*b^3*d*e^4 + 3*a^4*b^2*e^5)*x^8 + 1/7*(b^6*d^5 + 30*a*b^
5*d^4*e + 150*a^2*b^4*d^3*e^2 + 200*a^3*b^3*d^2*e^3 + 75*a^4*b^2*d*e^4 + 6*a^5*b*e^5)*x^7 + 1/6*(6*a*b^5*d^5 +
75*a^2*b^4*d^4*e + 200*a^3*b^3*d^3*e^2 + 150*a^4*b^2*d^2*e^3 + 30*a^5*b*d*e^4 + a^6*e^5)*x^6 + (3*a^2*b^4*d^5
+ 20*a^3*b^3*d^4*e + 30*a^4*b^2*d^3*e^2 + 12*a^5*b*d^2*e^3 + a^6*d*e^4)*x^5 + 5/4*(4*a^3*b^3*d^5 + 15*a^4*b^2
*d^4*e + 12*a^5*b*d^3*e^2 + 2*a^6*d^2*e^3)*x^4 + 5/3*(3*a^4*b^2*d^5 + 6*a^5*b*d^4*e + 2*a^6*d^3*e^2)*x^3 + 1/2
*(6*a^5*b*d^5 + 5*a^6*d^4*e)*x^2

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giac [B]  time = 0.21, size = 810, normalized size = 3.08 \begin {gather*} \frac {1}{12} \, b^{6} x^{12} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{11} \, b^{6} d x^{11} e^{4} \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{2} x^{10} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, b^{6} d^{3} x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, b^{6} d^{4} x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{11} \, a b^{5} x^{11} e^{5} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{5} d x^{10} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a b^{5} d^{2} x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a b^{5} d^{3} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {30}{7} \, a b^{5} d^{4} x^{7} e \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b^{4} x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{3} \, a^{2} b^{4} d x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {75}{4} \, a^{2} b^{4} d^{2} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {150}{7} \, a^{2} b^{4} d^{3} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{2} b^{4} d^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{9} \, a^{3} b^{3} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{3} b^{3} d x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {200}{7} \, a^{3} b^{3} d^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {100}{3} \, a^{3} b^{3} d^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{4} x^{5} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{8} \, a^{4} b^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {75}{7} \, a^{4} b^{2} d x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 25 \, a^{4} b^{2} d^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{4} b^{2} d^{3} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {75}{4} \, a^{4} b^{2} d^{4} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, a^{5} b x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{5} b d x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{5} b d^{2} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{5} b d^{3} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{5} b d^{4} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{6} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{6} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{6} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{6} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{5} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^6*x^12*e^5*sgn(b*x + a) + 5/11*b^6*d*x^11*e^4*sgn(b*x + a) + b^6*d^2*x^10*e^3*sgn(b*x + a) + 10/9*b^6*d
^3*x^9*e^2*sgn(b*x + a) + 5/8*b^6*d^4*x^8*e*sgn(b*x + a) + 1/7*b^6*d^5*x^7*sgn(b*x + a) + 6/11*a*b^5*x^11*e^5*
sgn(b*x + a) + 3*a*b^5*d*x^10*e^4*sgn(b*x + a) + 20/3*a*b^5*d^2*x^9*e^3*sgn(b*x + a) + 15/2*a*b^5*d^3*x^8*e^2*
sgn(b*x + a) + 30/7*a*b^5*d^4*x^7*e*sgn(b*x + a) + a*b^5*d^5*x^6*sgn(b*x + a) + 3/2*a^2*b^4*x^10*e^5*sgn(b*x +
a) + 25/3*a^2*b^4*d*x^9*e^4*sgn(b*x + a) + 75/4*a^2*b^4*d^2*x^8*e^3*sgn(b*x + a) + 150/7*a^2*b^4*d^3*x^7*e^2*
sgn(b*x + a) + 25/2*a^2*b^4*d^4*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^5*x^5*sgn(b*x + a) + 20/9*a^3*b^3*x^9*e^5*sgn
(b*x + a) + 25/2*a^3*b^3*d*x^8*e^4*sgn(b*x + a) + 200/7*a^3*b^3*d^2*x^7*e^3*sgn(b*x + a) + 100/3*a^3*b^3*d^3*x
^6*e^2*sgn(b*x + a) + 20*a^3*b^3*d^4*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^5*x^4*sgn(b*x + a) + 15/8*a^4*b^2*x^8*e^
5*sgn(b*x + a) + 75/7*a^4*b^2*d*x^7*e^4*sgn(b*x + a) + 25*a^4*b^2*d^2*x^6*e^3*sgn(b*x + a) + 30*a^4*b^2*d^3*x^
5*e^2*sgn(b*x + a) + 75/4*a^4*b^2*d^4*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^5*x^3*sgn(b*x + a) + 6/7*a^5*b*x^7*e^5*
sgn(b*x + a) + 5*a^5*b*d*x^6*e^4*sgn(b*x + a) + 12*a^5*b*d^2*x^5*e^3*sgn(b*x + a) + 15*a^5*b*d^3*x^4*e^2*sgn(b
*x + a) + 10*a^5*b*d^4*x^3*e*sgn(b*x + a) + 3*a^5*b*d^5*x^2*sgn(b*x + a) + 1/6*a^6*x^6*e^5*sgn(b*x + a) + a^6*
d*x^5*e^4*sgn(b*x + a) + 5/2*a^6*d^2*x^4*e^3*sgn(b*x + a) + 10/3*a^6*d^3*x^3*e^2*sgn(b*x + a) + 5/2*a^6*d^4*x^
2*e*sgn(b*x + a) + a^6*d^5*x*sgn(b*x + a)

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maple [B]  time = 0.06, size = 598, normalized size = 2.27 \begin {gather*} \frac {\left (462 b^{6} e^{5} x^{11}+3024 x^{10} e^{5} a \,b^{5}+2520 x^{10} d \,e^{4} b^{6}+8316 x^{9} e^{5} a^{2} b^{4}+16632 x^{9} d \,e^{4} a \,b^{5}+5544 x^{9} d^{2} e^{3} b^{6}+12320 x^{8} e^{5} a^{3} b^{3}+46200 x^{8} d \,e^{4} a^{2} b^{4}+36960 x^{8} d^{2} e^{3} a \,b^{5}+6160 x^{8} d^{3} e^{2} b^{6}+10395 x^{7} e^{5} a^{4} b^{2}+69300 x^{7} d \,e^{4} a^{3} b^{3}+103950 x^{7} d^{2} e^{3} a^{2} b^{4}+41580 x^{7} d^{3} e^{2} a \,b^{5}+3465 x^{7} d^{4} e \,b^{6}+4752 x^{6} e^{5} a^{5} b +59400 x^{6} d \,e^{4} a^{4} b^{2}+158400 x^{6} d^{2} e^{3} a^{3} b^{3}+118800 x^{6} d^{3} e^{2} a^{2} b^{4}+23760 x^{6} d^{4} e a \,b^{5}+792 x^{6} d^{5} b^{6}+924 x^{5} e^{5} a^{6}+27720 x^{5} d \,e^{4} a^{5} b +138600 x^{5} d^{2} e^{3} a^{4} b^{2}+184800 x^{5} d^{3} e^{2} a^{3} b^{3}+69300 x^{5} d^{4} e \,a^{2} b^{4}+5544 x^{5} d^{5} a \,b^{5}+5544 a^{6} d \,e^{4} x^{4}+66528 a^{5} b \,d^{2} e^{3} x^{4}+166320 a^{4} b^{2} d^{3} e^{2} x^{4}+110880 a^{3} b^{3} d^{4} e \,x^{4}+16632 a^{2} b^{4} d^{5} x^{4}+13860 x^{3} d^{2} e^{3} a^{6}+83160 x^{3} d^{3} e^{2} a^{5} b +103950 x^{3} d^{4} e \,a^{4} b^{2}+27720 x^{3} d^{5} a^{3} b^{3}+18480 x^{2} d^{3} e^{2} a^{6}+55440 x^{2} d^{4} e \,a^{5} b +27720 x^{2} d^{5} a^{4} b^{2}+13860 x \,d^{4} e \,a^{6}+16632 x \,d^{5} a^{5} b +5544 d^{5} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{5544 \left (b x +a \right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5544*x*(462*b^6*e^5*x^11+3024*a*b^5*e^5*x^10+2520*b^6*d*e^4*x^10+8316*a^2*b^4*e^5*x^9+16632*a*b^5*d*e^4*x^9+
5544*b^6*d^2*e^3*x^9+12320*a^3*b^3*e^5*x^8+46200*a^2*b^4*d*e^4*x^8+36960*a*b^5*d^2*e^3*x^8+6160*b^6*d^3*e^2*x^
8+10395*a^4*b^2*e^5*x^7+69300*a^3*b^3*d*e^4*x^7+103950*a^2*b^4*d^2*e^3*x^7+41580*a*b^5*d^3*e^2*x^7+3465*b^6*d^
4*e*x^7+4752*a^5*b*e^5*x^6+59400*a^4*b^2*d*e^4*x^6+158400*a^3*b^3*d^2*e^3*x^6+118800*a^2*b^4*d^3*e^2*x^6+23760
*a*b^5*d^4*e*x^6+792*b^6*d^5*x^6+924*a^6*e^5*x^5+27720*a^5*b*d*e^4*x^5+138600*a^4*b^2*d^2*e^3*x^5+184800*a^3*b
^3*d^3*e^2*x^5+69300*a^2*b^4*d^4*e*x^5+5544*a*b^5*d^5*x^5+5544*a^6*d*e^4*x^4+66528*a^5*b*d^2*e^3*x^4+166320*a^
4*b^2*d^3*e^2*x^4+110880*a^3*b^3*d^4*e*x^4+16632*a^2*b^4*d^5*x^4+13860*a^6*d^2*e^3*x^3+83160*a^5*b*d^3*e^2*x^3
+103950*a^4*b^2*d^4*e*x^3+27720*a^3*b^3*d^5*x^3+18480*a^6*d^3*e^2*x^2+55440*a^5*b*d^4*e*x^2+27720*a^4*b^2*d^5*
x^2+13860*a^6*d^4*e*x+16632*a^5*b*d^5*x+5544*a^6*d^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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maxima [B]  time = 0.64, size = 1323, normalized size = 5.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^5*x^5/b - 17/132*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^5*x^4/b^2 + 5/33*(
b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^5*x^3/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^5*x + 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*a^6*e^5*x/b^5 - 16/99*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^5*x^2/b^4 + 1/6*(b^2*x^2 + 2*
a*b*x + a^2)^(5/2)*a^2*d^5/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7*e^5/b^6 + 131/792*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*a^4*e^5*x/b^5 - 923/5544*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*e^5/b^6 + 1/11*(5*b*d*e^4 + a*e^5)*(b^
2*x^2 + 2*a*b*x + a^2)^(7/2)*x^4/b^2 - 3/22*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^3/b^3 + 1/
2*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 - 1/6*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a^5*x/b^5 + 5/6*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/b^4 - 5/3*(b*d^3*e
^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 5/6*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a
^2)^(5/2)*a^2*x/b^2 - 1/6*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b + 31/198*(5*b*d*e^4 + a*e^
5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^2/b^4 - 13/18*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)
*a*x^2/b^3 + 10/9*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^2/b^2 - 1/6*(5*b*d*e^4 + a*e^5)*(b
^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6/b^6 + 5/6*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b^5 -
5/3*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 5/6*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 - 65/396*(5*b*d*
e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x/b^5 + 29/36*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^
2)^(7/2)*a^2*x/b^4 - 55/36*(b*d^3*e^2 + a*d^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 5/8*(b*d^4*e + 2*
a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x/b^2 + 461/2772*(5*b*d*e^4 + a*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2
)*a^4/b^6 - 209/252*(2*b*d^2*e^3 + a*d*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5 + 415/252*(b*d^3*e^2 + a*d
^2*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 45/56*(b*d^4*e + 2*a*d^3*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2
)*a/b^3 + 1/7*(b*d^5 + 5*a*d^4*e)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/b^2

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

[Out]

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

[Out]

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

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