3.18.68 \(\int (a+b x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=219 \[ \frac {2 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{5 b^5}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{3 b^5}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{2 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^5} \]

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Rubi [A]  time = 0.30, antiderivative size = 219, 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 {2 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)}{5 b^5}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2}{3 b^5}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3}{2 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4}{7 b^5}+\frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.12, size = 371, normalized size = 1.69 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (462 a^6 \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+462 a^5 b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+330 a^4 b^2 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+165 a^3 b^3 x^3 \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+55 a^2 b^4 x^4 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+11 a b^5 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )+b^6 x^6 \left (330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right )\right )}{2310 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(462*a^6*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + 462*a^5*b*x*(15*
d^4 + 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 330*a^4*b^2*x^2*(35*d^4 + 105*d^3*e*x + 126*d^
2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 165*a^3*b^3*x^3*(70*d^4 + 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x
^3 + 35*e^4*x^4) + 55*a^2*b^4*x^4*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + 11*
a*b^5*x^5*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + b^6*x^6*(330*d^4 + 1155*d^
3*e*x + 1540*d^2*e^2*x^2 + 924*d*e^3*x^3 + 210*e^4*x^4)))/(2310*(a + b*x))

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IntegrateAlgebraic [F]  time = 3.44, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^4 \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)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(5/2), x]

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fricas [B]  time = 0.42, size = 418, normalized size = 1.91 \begin {gather*} \frac {1}{11} \, b^{6} e^{4} x^{11} + a^{6} d^{4} x + \frac {1}{5} \, {\left (2 \, b^{6} d e^{3} + 3 \, a b^{5} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, b^{6} d^{2} e^{2} + 8 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (b^{6} d^{3} e + 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} + 5 \, a^{3} b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{4} + 24 \, a b^{5} d^{3} e + 90 \, a^{2} b^{4} d^{2} e^{2} + 80 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x^{7} + {\left (a b^{5} d^{4} + 10 \, a^{2} b^{4} d^{3} e + 20 \, a^{3} b^{3} d^{2} e^{2} + 10 \, a^{4} b^{2} d e^{3} + a^{5} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{4} + 80 \, a^{3} b^{3} d^{3} e + 90 \, a^{4} b^{2} d^{2} e^{2} + 24 \, a^{5} b d e^{3} + a^{6} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{4} + 15 \, a^{4} b^{2} d^{3} e + 9 \, a^{5} b d^{2} e^{2} + a^{6} d e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{4} + 8 \, a^{5} b d^{3} e + 2 \, a^{6} d^{2} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{4} + 2 \, a^{6} d^{3} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/11*b^6*e^4*x^11 + a^6*d^4*x + 1/5*(2*b^6*d*e^3 + 3*a*b^5*e^4)*x^10 + 1/3*(2*b^6*d^2*e^2 + 8*a*b^5*d*e^3 + 5*
a^2*b^4*e^4)*x^9 + 1/2*(b^6*d^3*e + 9*a*b^5*d^2*e^2 + 15*a^2*b^4*d*e^3 + 5*a^3*b^3*e^4)*x^8 + 1/7*(b^6*d^4 + 2
4*a*b^5*d^3*e + 90*a^2*b^4*d^2*e^2 + 80*a^3*b^3*d*e^3 + 15*a^4*b^2*e^4)*x^7 + (a*b^5*d^4 + 10*a^2*b^4*d^3*e +
20*a^3*b^3*d^2*e^2 + 10*a^4*b^2*d*e^3 + a^5*b*e^4)*x^6 + 1/5*(15*a^2*b^4*d^4 + 80*a^3*b^3*d^3*e + 90*a^4*b^2*d
^2*e^2 + 24*a^5*b*d*e^3 + a^6*e^4)*x^5 + (5*a^3*b^3*d^4 + 15*a^4*b^2*d^3*e + 9*a^5*b*d^2*e^2 + a^6*d*e^3)*x^4
+ (5*a^4*b^2*d^4 + 8*a^5*b*d^3*e + 2*a^6*d^2*e^2)*x^3 + (3*a^5*b*d^4 + 2*a^6*d^3*e)*x^2

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giac [B]  time = 0.19, size = 666, normalized size = 3.04 \begin {gather*} \frac {1}{11} \, b^{6} x^{11} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, b^{6} d x^{10} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, b^{6} d^{2} x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b^{6} d^{3} x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{4} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, a b^{5} x^{10} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {8}{3} \, a b^{5} d x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a b^{5} d^{2} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {24}{7} \, a b^{5} d^{3} x^{7} e \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{4} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{2} b^{4} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b^{4} d x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {90}{7} \, a^{2} b^{4} d^{2} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} b^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {80}{7} \, a^{3} b^{3} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 16 \, a^{3} b^{3} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{7} \, a^{4} b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{4} b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 18 \, a^{4} b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{5} b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {24}{5} \, a^{5} b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 9 \, a^{5} b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, a^{5} b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, a^{6} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{6} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{6} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{6} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{4} 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)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/11*b^6*x^11*e^4*sgn(b*x + a) + 2/5*b^6*d*x^10*e^3*sgn(b*x + a) + 2/3*b^6*d^2*x^9*e^2*sgn(b*x + a) + 1/2*b^6*
d^3*x^8*e*sgn(b*x + a) + 1/7*b^6*d^4*x^7*sgn(b*x + a) + 3/5*a*b^5*x^10*e^4*sgn(b*x + a) + 8/3*a*b^5*d*x^9*e^3*
sgn(b*x + a) + 9/2*a*b^5*d^2*x^8*e^2*sgn(b*x + a) + 24/7*a*b^5*d^3*x^7*e*sgn(b*x + a) + a*b^5*d^4*x^6*sgn(b*x
+ a) + 5/3*a^2*b^4*x^9*e^4*sgn(b*x + a) + 15/2*a^2*b^4*d*x^8*e^3*sgn(b*x + a) + 90/7*a^2*b^4*d^2*x^7*e^2*sgn(b
*x + a) + 10*a^2*b^4*d^3*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^4*x^5*sgn(b*x + a) + 5/2*a^3*b^3*x^8*e^4*sgn(b*x + a
) + 80/7*a^3*b^3*d*x^7*e^3*sgn(b*x + a) + 20*a^3*b^3*d^2*x^6*e^2*sgn(b*x + a) + 16*a^3*b^3*d^3*x^5*e*sgn(b*x +
a) + 5*a^3*b^3*d^4*x^4*sgn(b*x + a) + 15/7*a^4*b^2*x^7*e^4*sgn(b*x + a) + 10*a^4*b^2*d*x^6*e^3*sgn(b*x + a) +
18*a^4*b^2*d^2*x^5*e^2*sgn(b*x + a) + 15*a^4*b^2*d^3*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^4*x^3*sgn(b*x + a) + a^
5*b*x^6*e^4*sgn(b*x + a) + 24/5*a^5*b*d*x^5*e^3*sgn(b*x + a) + 9*a^5*b*d^2*x^4*e^2*sgn(b*x + a) + 8*a^5*b*d^3*
x^3*e*sgn(b*x + a) + 3*a^5*b*d^4*x^2*sgn(b*x + a) + 1/5*a^6*x^5*e^4*sgn(b*x + a) + a^6*d*x^4*e^3*sgn(b*x + a)
+ 2*a^6*d^2*x^3*e^2*sgn(b*x + a) + 2*a^6*d^3*x^2*e*sgn(b*x + a) + a^6*d^4*x*sgn(b*x + a)

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maple [B]  time = 0.06, size = 489, normalized size = 2.23 \begin {gather*} \frac {\left (210 e^{4} b^{6} x^{10}+1386 x^{9} e^{4} a \,b^{5}+924 x^{9} d \,e^{3} b^{6}+3850 x^{8} e^{4} a^{2} b^{4}+6160 x^{8} d \,e^{3} a \,b^{5}+1540 x^{8} d^{2} e^{2} b^{6}+5775 x^{7} e^{4} a^{3} b^{3}+17325 x^{7} d \,e^{3} a^{2} b^{4}+10395 x^{7} d^{2} e^{2} a \,b^{5}+1155 x^{7} d^{3} e \,b^{6}+4950 x^{6} e^{4} a^{4} b^{2}+26400 x^{6} d \,e^{3} a^{3} b^{3}+29700 x^{6} d^{2} e^{2} a^{2} b^{4}+7920 x^{6} d^{3} e a \,b^{5}+330 x^{6} d^{4} b^{6}+2310 a^{5} b \,e^{4} x^{5}+23100 a^{4} b^{2} d \,e^{3} x^{5}+46200 a^{3} b^{3} d^{2} e^{2} x^{5}+23100 a^{2} b^{4} d^{3} e \,x^{5}+2310 a \,b^{5} d^{4} x^{5}+462 x^{4} e^{4} a^{6}+11088 x^{4} d \,e^{3} a^{5} b +41580 x^{4} d^{2} e^{2} a^{4} b^{2}+36960 x^{4} d^{3} e \,a^{3} b^{3}+6930 x^{4} d^{4} a^{2} b^{4}+2310 a^{6} d \,e^{3} x^{3}+20790 a^{5} b \,d^{2} e^{2} x^{3}+34650 a^{4} b^{2} d^{3} e \,x^{3}+11550 a^{3} b^{3} d^{4} x^{3}+4620 a^{6} d^{2} e^{2} x^{2}+18480 a^{5} b \,d^{3} e \,x^{2}+11550 a^{4} b^{2} d^{4} x^{2}+4620 a^{6} d^{3} e x +6930 a^{5} b \,d^{4} x +2310 d^{4} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{2310 \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)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/2310*x*(210*b^6*e^4*x^10+1386*a*b^5*e^4*x^9+924*b^6*d*e^3*x^9+3850*a^2*b^4*e^4*x^8+6160*a*b^5*d*e^3*x^8+1540
*b^6*d^2*e^2*x^8+5775*a^3*b^3*e^4*x^7+17325*a^2*b^4*d*e^3*x^7+10395*a*b^5*d^2*e^2*x^7+1155*b^6*d^3*e*x^7+4950*
a^4*b^2*e^4*x^6+26400*a^3*b^3*d*e^3*x^6+29700*a^2*b^4*d^2*e^2*x^6+7920*a*b^5*d^3*e*x^6+330*b^6*d^4*x^6+2310*a^
5*b*e^4*x^5+23100*a^4*b^2*d*e^3*x^5+46200*a^3*b^3*d^2*e^2*x^5+23100*a^2*b^4*d^3*e*x^5+2310*a*b^5*d^4*x^5+462*a
^6*e^4*x^4+11088*a^5*b*d*e^3*x^4+41580*a^4*b^2*d^2*e^2*x^4+36960*a^3*b^3*d^3*e*x^4+6930*a^2*b^4*d^4*x^4+2310*a
^6*d*e^3*x^3+20790*a^5*b*d^2*e^2*x^3+34650*a^4*b^2*d^3*e*x^3+11550*a^3*b^3*d^4*x^3+4620*a^6*d^2*e^2*x^2+18480*
a^5*b*d^3*e*x^2+11550*a^4*b^2*d^4*x^2+4620*a^6*d^3*e*x+6930*a^5*b*d^4*x+2310*a^6*d^4)*((b*x+a)^2)^(5/2)/(b*x+a
)^5

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maxima [B]  time = 0.53, size = 998, normalized size = 4.56

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^4*(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^4*x^4/b - 3/22*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^4*x^3/b^2 + 1/6*(b^2
*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^4*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^4*x/b^4 + 31/198*(b^2*x^2 + 2*
a*b*x + a^2)^(7/2)*a^2*e^4*x^2/b^3 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^4/b - 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*a^6*e^4/b^5 - 65/396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^4*x/b^4 + 461/2772*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*a^4*e^4/b^5 + 1/10*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^3/b^2 + 1/6*(4*b*d*e^3 + a
*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*x/b^4 - 1/3*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2
)*a^3*x/b^3 + 1/3*(2*b*d^3*e + 3*a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(b*d^4 + 4*a*d^3*e
)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b - 13/90*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^2/b^3
+ 2/9*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^2/b^2 + 1/6*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*a^5/b^5 - 1/3*(3*b*d^2*e^2 + 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 1/3*(2*b
*d^3*e + 3*a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(b*d^4 + 4*a*d^3*e)*(b^2*x^2 + 2*a*b*x + a
^2)^(5/2)*a^2/b^2 + 29/180*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x/b^4 - 11/36*(3*b*d^2*e^2
+ 2*a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 1/4*(2*b*d^3*e + 3*a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)
^(7/2)*x/b^2 - 209/1260*(4*b*d*e^3 + a*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3/b^5 + 83/252*(3*b*d^2*e^2 + 2*
a*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 9/28*(2*b*d^3*e + 3*a*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7
/2)*a/b^3 + 1/7*(b*d^4 + 4*a*d^3*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 )}^4\,{\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)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((a + b*x)*(d + e*x)^4*(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 )^{4} \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)**4*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((a + b*x)*(d + e*x)**4*((a + b*x)**2)**(5/2), x)

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