3.1740 \(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=324 \[ \frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (-5 a B e+A b e+4 b B d)}{10 b^6}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{9 b^6}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{4 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{7 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^4}{6 b^6}+\frac {B e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]

[Out]

1/6*(A*b-B*a)*(-a*e+b*d)^4*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^6+1/7*(-a*e+b*d)^3*(4*A*b*e-5*B*a*e+B*b*d)*(b*x+a)^6*
((b*x+a)^2)^(1/2)/b^6+1/4*e*(-a*e+b*d)^2*(3*A*b*e-5*B*a*e+2*B*b*d)*(b*x+a)^7*((b*x+a)^2)^(1/2)/b^6+2/9*e^2*(-a
*e+b*d)*(2*A*b*e-5*B*a*e+3*B*b*d)*(b*x+a)^8*((b*x+a)^2)^(1/2)/b^6+1/10*e^3*(A*b*e-5*B*a*e+4*B*b*d)*(b*x+a)^9*(
(b*x+a)^2)^(1/2)/b^6+1/11*B*e^4*(b*x+a)^10*((b*x+a)^2)^(1/2)/b^6

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Rubi [A]  time = 0.62, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (-5 a B e+A b e+4 b B d)}{10 b^6}+\frac {2 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{9 b^6}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{4 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{7 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^4}{6 b^6}+\frac {B e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]

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]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 \left (a b+b^2 x\right )^5 (A+B x) (d+e x)^4 \, 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 {(A b-a B) (b d-a e)^4 \left (a b+b^2 x\right )^5}{b^5}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) \left (a b+b^2 x\right )^6}{b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \left (a b+b^2 x\right )^7}{b^7}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) \left (a b+b^2 x\right )^8}{b^8}+\frac {e^3 (4 b B d+A b e-5 a B e) \left (a b+b^2 x\right )^9}{b^9}+\frac {B e^4 \left (a b+b^2 x\right )^{10}}{b^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e)^4 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac {e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{4 b^6}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^6}+\frac {B e^4 (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 = 0.24, size = 611, normalized size = 1.89 \[ \frac {x \sqrt {(a+b x)^2} \left (462 a^5 \left (6 A \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+B x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )+330 a^4 b x \left (7 A \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )+2 B x \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )\right )+165 a^3 b^2 x^2 \left (8 A \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+3 B x \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )\right )+55 a^2 b^3 x^3 \left (9 A \left (70 d^4+224 d^3 e x+280 d^2 e^2 x^2+160 d e^3 x^3+35 e^4 x^4\right )+4 B x \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )\right )+55 a b^4 x^4 \left (2 A \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+B x \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )\right )+b^5 x^5 \left (11 A \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )+6 B x \left (330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right )\right )\right )}{13860 (a+b x)} \]

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^5*(6*A*(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) + 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*x*(7*A*(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) + 2*B*x*(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^2*x^2*(8*A*(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) + 3*B*x*(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^3*x^3*(9*A*(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) + 4*B*x*(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)) + 55*a*b^4*x^4*(2*A*(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) + B
*x*(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^5*x^5*(11*A*(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) + 6*B*x*(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))))/(13860*(a + b*x))

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fricas [B]  time = 0.83, size = 677, normalized size = 2.09 \[ \frac {1}{11} \, B b^{5} e^{4} x^{11} + A a^{5} d^{4} x + \frac {1}{10} \, {\left (4 \, B b^{5} d e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (6 \, B b^{5} d^{2} e^{2} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{3} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{4}\right )} x^{9} + \frac {1}{4} \, {\left (2 \, B b^{5} d^{3} e + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{3} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{4} + 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{2} + 40 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{3} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} + 20 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e + 60 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{2} + 20 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{5} e^{4} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} + 40 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, A a^{5} d e^{3} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{5} d^{2} e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{5} d^{3} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4}\right )} x^{2} \]

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*b^5*e^4*x^11 + A*a^5*d^4*x + 1/10*(4*B*b^5*d*e^3 + (5*B*a*b^4 + A*b^5)*e^4)*x^10 + 1/9*(6*B*b^5*d^2*e^2
+ 4*(5*B*a*b^4 + A*b^5)*d*e^3 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^4)*x^9 + 1/4*(2*B*b^5*d^3*e + 3*(5*B*a*b^4 + A*b^
5)*d^2*e^2 + 10*(2*B*a^2*b^3 + A*a*b^4)*d*e^3 + 5*(B*a^3*b^2 + A*a^2*b^3)*e^4)*x^8 + 1/7*(B*b^5*d^4 + 4*(5*B*a
*b^4 + A*b^5)*d^3*e + 30*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^2 + 40*(B*a^3*b^2 + A*a^2*b^3)*d*e^3 + 5*(B*a^4*b + 2*A
*a^3*b^2)*e^4)*x^7 + 1/6*((5*B*a*b^4 + A*b^5)*d^4 + 20*(2*B*a^2*b^3 + A*a*b^4)*d^3*e + 60*(B*a^3*b^2 + A*a^2*b
^3)*d^2*e^2 + 20*(B*a^4*b + 2*A*a^3*b^2)*d*e^3 + (B*a^5 + 5*A*a^4*b)*e^4)*x^6 + 1/5*(A*a^5*e^4 + 5*(2*B*a^2*b^
3 + A*a*b^4)*d^4 + 40*(B*a^3*b^2 + A*a^2*b^3)*d^3*e + 30*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^2 + 4*(B*a^5 + 5*A*a^4*
b)*d*e^3)*x^5 + 1/2*(2*A*a^5*d*e^3 + 5*(B*a^3*b^2 + A*a^2*b^3)*d^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d^3*e + 3*(B*a
^5 + 5*A*a^4*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^5*d^2*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^4 + 4*(B*a^5 + 5*A*a^4*b)*d^
3*e)*x^3 + 1/2*(4*A*a^5*d^3*e + (B*a^5 + 5*A*a^4*b)*d^4)*x^2

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giac [B]  time = 0.25, size = 1192, normalized size = 3.68 \[ \text {result too large to display} \]

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="giac")

[Out]

1/11*B*b^5*x^11*e^4*sgn(b*x + a) + 2/5*B*b^5*d*x^10*e^3*sgn(b*x + a) + 2/3*B*b^5*d^2*x^9*e^2*sgn(b*x + a) + 1/
2*B*b^5*d^3*x^8*e*sgn(b*x + a) + 1/7*B*b^5*d^4*x^7*sgn(b*x + a) + 1/2*B*a*b^4*x^10*e^4*sgn(b*x + a) + 1/10*A*b
^5*x^10*e^4*sgn(b*x + a) + 20/9*B*a*b^4*d*x^9*e^3*sgn(b*x + a) + 4/9*A*b^5*d*x^9*e^3*sgn(b*x + a) + 15/4*B*a*b
^4*d^2*x^8*e^2*sgn(b*x + a) + 3/4*A*b^5*d^2*x^8*e^2*sgn(b*x + a) + 20/7*B*a*b^4*d^3*x^7*e*sgn(b*x + a) + 4/7*A
*b^5*d^3*x^7*e*sgn(b*x + a) + 5/6*B*a*b^4*d^4*x^6*sgn(b*x + a) + 1/6*A*b^5*d^4*x^6*sgn(b*x + a) + 10/9*B*a^2*b
^3*x^9*e^4*sgn(b*x + a) + 5/9*A*a*b^4*x^9*e^4*sgn(b*x + a) + 5*B*a^2*b^3*d*x^8*e^3*sgn(b*x + a) + 5/2*A*a*b^4*
d*x^8*e^3*sgn(b*x + a) + 60/7*B*a^2*b^3*d^2*x^7*e^2*sgn(b*x + a) + 30/7*A*a*b^4*d^2*x^7*e^2*sgn(b*x + a) + 20/
3*B*a^2*b^3*d^3*x^6*e*sgn(b*x + a) + 10/3*A*a*b^4*d^3*x^6*e*sgn(b*x + a) + 2*B*a^2*b^3*d^4*x^5*sgn(b*x + a) +
A*a*b^4*d^4*x^5*sgn(b*x + a) + 5/4*B*a^3*b^2*x^8*e^4*sgn(b*x + a) + 5/4*A*a^2*b^3*x^8*e^4*sgn(b*x + a) + 40/7*
B*a^3*b^2*d*x^7*e^3*sgn(b*x + a) + 40/7*A*a^2*b^3*d*x^7*e^3*sgn(b*x + a) + 10*B*a^3*b^2*d^2*x^6*e^2*sgn(b*x +
a) + 10*A*a^2*b^3*d^2*x^6*e^2*sgn(b*x + a) + 8*B*a^3*b^2*d^3*x^5*e*sgn(b*x + a) + 8*A*a^2*b^3*d^3*x^5*e*sgn(b*
x + a) + 5/2*B*a^3*b^2*d^4*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d^4*x^4*sgn(b*x + a) + 5/7*B*a^4*b*x^7*e^4*sgn(b*x
+ a) + 10/7*A*a^3*b^2*x^7*e^4*sgn(b*x + a) + 10/3*B*a^4*b*d*x^6*e^3*sgn(b*x + a) + 20/3*A*a^3*b^2*d*x^6*e^3*s
gn(b*x + a) + 6*B*a^4*b*d^2*x^5*e^2*sgn(b*x + a) + 12*A*a^3*b^2*d^2*x^5*e^2*sgn(b*x + a) + 5*B*a^4*b*d^3*x^4*e
*sgn(b*x + a) + 10*A*a^3*b^2*d^3*x^4*e*sgn(b*x + a) + 5/3*B*a^4*b*d^4*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^4*x^
3*sgn(b*x + a) + 1/6*B*a^5*x^6*e^4*sgn(b*x + a) + 5/6*A*a^4*b*x^6*e^4*sgn(b*x + a) + 4/5*B*a^5*d*x^5*e^3*sgn(b
*x + a) + 4*A*a^4*b*d*x^5*e^3*sgn(b*x + a) + 3/2*B*a^5*d^2*x^4*e^2*sgn(b*x + a) + 15/2*A*a^4*b*d^2*x^4*e^2*sgn
(b*x + a) + 4/3*B*a^5*d^3*x^3*e*sgn(b*x + a) + 20/3*A*a^4*b*d^3*x^3*e*sgn(b*x + a) + 1/2*B*a^5*d^4*x^2*sgn(b*x
+ a) + 5/2*A*a^4*b*d^4*x^2*sgn(b*x + a) + 1/5*A*a^5*x^5*e^4*sgn(b*x + a) + A*a^5*d*x^4*e^3*sgn(b*x + a) + 2*A
*a^5*d^2*x^3*e^2*sgn(b*x + a) + 2*A*a^5*d^3*x^2*e*sgn(b*x + a) + A*a^5*d^4*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 872, normalized size = 2.69 \[ \frac {\left (1260 B \,b^{5} e^{4} x^{10}+1386 x^{9} A \,b^{5} e^{4}+6930 x^{9} B \,e^{4} a \,b^{4}+5544 x^{9} B \,b^{5} d \,e^{3}+7700 x^{8} A a \,b^{4} e^{4}+6160 x^{8} A \,b^{5} d \,e^{3}+15400 x^{8} B \,e^{4} a^{2} b^{3}+30800 x^{8} B a \,b^{4} d \,e^{3}+9240 x^{8} B \,b^{5} d^{2} e^{2}+17325 x^{7} A \,a^{2} b^{3} e^{4}+34650 x^{7} A a \,b^{4} d \,e^{3}+10395 x^{7} A \,b^{5} d^{2} e^{2}+17325 x^{7} B \,e^{4} a^{3} b^{2}+69300 x^{7} B \,a^{2} b^{3} d \,e^{3}+51975 x^{7} B a \,b^{4} d^{2} e^{2}+6930 x^{7} B \,b^{5} d^{3} e +19800 x^{6} A \,a^{3} b^{2} e^{4}+79200 x^{6} A \,a^{2} b^{3} d \,e^{3}+59400 x^{6} A a \,b^{4} d^{2} e^{2}+7920 x^{6} A \,b^{5} d^{3} e +9900 x^{6} B \,e^{4} a^{4} b +79200 x^{6} B \,a^{3} b^{2} d \,e^{3}+118800 x^{6} B \,a^{2} b^{3} d^{2} e^{2}+39600 x^{6} B a \,b^{4} d^{3} e +1980 x^{6} B \,b^{5} d^{4}+11550 x^{5} A \,a^{4} b \,e^{4}+92400 x^{5} A \,a^{3} b^{2} d \,e^{3}+138600 x^{5} A \,a^{2} b^{3} d^{2} e^{2}+46200 x^{5} A a \,b^{4} d^{3} e +2310 x^{5} A \,d^{4} b^{5}+2310 x^{5} B \,e^{4} a^{5}+46200 x^{5} B \,a^{4} b d \,e^{3}+138600 x^{5} B \,a^{3} b^{2} d^{2} e^{2}+92400 x^{5} B \,a^{2} b^{3} d^{3} e +11550 x^{5} B a \,b^{4} d^{4}+2772 x^{4} A \,a^{5} e^{4}+55440 x^{4} A \,a^{4} b d \,e^{3}+166320 x^{4} A \,a^{3} b^{2} d^{2} e^{2}+110880 x^{4} A \,a^{2} b^{3} d^{3} e +13860 x^{4} A \,d^{4} a \,b^{4}+11088 x^{4} B \,a^{5} d \,e^{3}+83160 x^{4} B \,a^{4} b \,d^{2} e^{2}+110880 x^{4} B \,a^{3} b^{2} d^{3} e +27720 x^{4} B \,a^{2} b^{3} d^{4}+13860 x^{3} A \,a^{5} d \,e^{3}+103950 x^{3} A \,a^{4} b \,d^{2} e^{2}+138600 x^{3} A \,a^{3} b^{2} d^{3} e +34650 x^{3} A \,d^{4} a^{2} b^{3}+20790 x^{3} B \,a^{5} d^{2} e^{2}+69300 x^{3} B \,a^{4} b \,d^{3} e +34650 x^{3} B \,a^{3} b^{2} d^{4}+27720 x^{2} A \,a^{5} d^{2} e^{2}+92400 x^{2} A \,a^{4} b \,d^{3} e +46200 x^{2} A \,d^{4} a^{3} b^{2}+18480 x^{2} B \,a^{5} d^{3} e +23100 x^{2} B \,a^{4} b \,d^{4}+27720 x A \,a^{5} d^{3} e +34650 x A \,d^{4} a^{4} b +6930 x B \,a^{5} d^{4}+13860 A \,d^{4} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{13860 \left (b x +a \right )^{5}} \]

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]

1/13860*x*(1260*B*b^5*e^4*x^10+1386*A*b^5*e^4*x^9+6930*B*a*b^4*e^4*x^9+5544*B*b^5*d*e^3*x^9+7700*A*a*b^4*e^4*x
^8+6160*A*b^5*d*e^3*x^8+15400*B*a^2*b^3*e^4*x^8+30800*B*a*b^4*d*e^3*x^8+9240*B*b^5*d^2*e^2*x^8+17325*A*a^2*b^3
*e^4*x^7+34650*A*a*b^4*d*e^3*x^7+10395*A*b^5*d^2*e^2*x^7+17325*B*a^3*b^2*e^4*x^7+69300*B*a^2*b^3*d*e^3*x^7+519
75*B*a*b^4*d^2*e^2*x^7+6930*B*b^5*d^3*e*x^7+19800*A*a^3*b^2*e^4*x^6+79200*A*a^2*b^3*d*e^3*x^6+59400*A*a*b^4*d^
2*e^2*x^6+7920*A*b^5*d^3*e*x^6+9900*B*a^4*b*e^4*x^6+79200*B*a^3*b^2*d*e^3*x^6+118800*B*a^2*b^3*d^2*e^2*x^6+396
00*B*a*b^4*d^3*e*x^6+1980*B*b^5*d^4*x^6+11550*A*a^4*b*e^4*x^5+92400*A*a^3*b^2*d*e^3*x^5+138600*A*a^2*b^3*d^2*e
^2*x^5+46200*A*a*b^4*d^3*e*x^5+2310*A*b^5*d^4*x^5+2310*B*a^5*e^4*x^5+46200*B*a^4*b*d*e^3*x^5+138600*B*a^3*b^2*
d^2*e^2*x^5+92400*B*a^2*b^3*d^3*e*x^5+11550*B*a*b^4*d^4*x^5+2772*A*a^5*e^4*x^4+55440*A*a^4*b*d*e^3*x^4+166320*
A*a^3*b^2*d^2*e^2*x^4+110880*A*a^2*b^3*d^3*e*x^4+13860*A*a*b^4*d^4*x^4+11088*B*a^5*d*e^3*x^4+83160*B*a^4*b*d^2
*e^2*x^4+110880*B*a^3*b^2*d^3*e*x^4+27720*B*a^2*b^3*d^4*x^4+13860*A*a^5*d*e^3*x^3+103950*A*a^4*b*d^2*e^2*x^3+1
38600*A*a^3*b^2*d^3*e*x^3+34650*A*a^2*b^3*d^4*x^3+20790*B*a^5*d^2*e^2*x^3+69300*B*a^4*b*d^3*e*x^3+34650*B*a^3*
b^2*d^4*x^3+27720*A*a^5*d^2*e^2*x^2+92400*A*a^4*b*d^3*e*x^2+46200*A*a^3*b^2*d^4*x^2+18480*B*a^5*d^3*e*x^2+2310
0*B*a^4*b*d^4*x^2+27720*A*a^5*d^3*e*x+34650*A*a^4*b*d^4*x+6930*B*a^5*d^4*x+13860*A*a^5*d^4)*((b*x+a)^2)^(5/2)/
(b*x+a)^5

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maxima [B]  time = 0.65, size = 1004, normalized size = 3.10 \[ \text {result too large to display} \]

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)*B*e^4*x^4/b^2 - 3/22*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*e^4*x^3/b^3 + 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)*B*a^5*e^4*x/b^5 + 31/198*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*B*a^2*e^4*x^2/b^4 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*a*d^4/b - 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*B*a^6*e^4/b^6 - 65/396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*e^4*x/b^5 + 461/2772*(b^2*x^
2 + 2*a*b*x + a^2)^(7/2)*B*a^4*e^4/b^6 + 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 \[ \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^4\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

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 \[ \int \left (A + B x\right ) \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]

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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