3.16.1 \(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=298 \[ -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^5 (a+b x)} \]

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Rubi [A]  time = 0.40, antiderivative size = 298, 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} \begin {gather*} -\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x)}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9}{9 e^5 (a+b x)} \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)^(3/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.16, size = 410, normalized size = 1.38 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (84 a^3 \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 )+36 a^2 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 )+9 a 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 )+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 )\right )}{2520 (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)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(84*a^3*(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)) + 36*a^2*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))
+ 9*a*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)) + 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
))))/(2520*(a + b*x))

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IntegrateAlgebraic [F]  time = 5.33, 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 )^{3/2} \, 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)^(3/2),x]

[Out]

Defer[IntegrateAlgebraic][(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)^(3/2), x]

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fricas [A]  time = 0.42, size = 425, normalized size = 1.43 \begin {gather*} \frac {1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{3} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, B b^{3} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\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)^(3/2),x, algorithm="fricas")

[Out]

1/9*B*b^3*e^4*x^9 + A*a^3*d^4*x + 1/8*(4*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^8 + 1/7*(6*B*b^3*d^2*e^2 + 4
*(3*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^7 + 1/6*(4*B*b^3*d^3*e + 6*(3*B*a*b^2 + A*b^3)*d^2*e
^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x^6 + 1/5*(B*b^3*d^4 + A*a^3*e^4 + 4*(3*B*a*b^2 +
A*b^3)*d^3*e + 18*(B*a^2*b + A*a*b^2)*d^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A*a^3*d*e^3 + (3*B*
a*b^2 + A*b^3)*d^4 + 12*(B*a^2*b + A*a*b^2)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^3*d^2*e^2
+ 3*(B*a^2*b + A*a*b^2)*d^4 + 4*(B*a^3 + 3*A*a^2*b)*d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)
*x^2

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giac [B]  time = 0.20, size = 758, normalized size = 2.54 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B b^{3} d x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, B b^{3} d^{2} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B b^{3} d^{3} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B b^{3} d^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, B a b^{2} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{8} \, A b^{3} x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{7} \, B a b^{2} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{7} \, A b^{3} d x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a b^{2} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + A b^{3} d^{2} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, B a b^{2} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, A b^{3} d^{3} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a b^{2} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A b^{3} d^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, B a^{2} b x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, A a b^{2} x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{2} b d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a b^{2} d x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, B a^{2} b d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{5} \, A a b^{2} d^{2} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a^{2} b d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a b^{2} d^{3} x^{4} e \mathrm {sgn}\left (b x + a\right ) + B a^{2} b d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} d^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, B a^{3} x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a^{2} b x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{5} \, B a^{3} d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {12}{5} \, A a^{2} b d x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a^{3} d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, A a^{2} b d^{2} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {4}{3} \, B a^{3} d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{2} b d^{3} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{3} d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{2} b d^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, A a^{3} x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + A a^{3} d x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} d^{2} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} d^{3} x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a^{3} 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)^(3/2),x, algorithm="giac")

[Out]

1/9*B*b^3*x^9*e^4*sgn(b*x + a) + 1/2*B*b^3*d*x^8*e^3*sgn(b*x + a) + 6/7*B*b^3*d^2*x^7*e^2*sgn(b*x + a) + 2/3*B
*b^3*d^3*x^6*e*sgn(b*x + a) + 1/5*B*b^3*d^4*x^5*sgn(b*x + a) + 3/8*B*a*b^2*x^8*e^4*sgn(b*x + a) + 1/8*A*b^3*x^
8*e^4*sgn(b*x + a) + 12/7*B*a*b^2*d*x^7*e^3*sgn(b*x + a) + 4/7*A*b^3*d*x^7*e^3*sgn(b*x + a) + 3*B*a*b^2*d^2*x^
6*e^2*sgn(b*x + a) + A*b^3*d^2*x^6*e^2*sgn(b*x + a) + 12/5*B*a*b^2*d^3*x^5*e*sgn(b*x + a) + 4/5*A*b^3*d^3*x^5*
e*sgn(b*x + a) + 3/4*B*a*b^2*d^4*x^4*sgn(b*x + a) + 1/4*A*b^3*d^4*x^4*sgn(b*x + a) + 3/7*B*a^2*b*x^7*e^4*sgn(b
*x + a) + 3/7*A*a*b^2*x^7*e^4*sgn(b*x + a) + 2*B*a^2*b*d*x^6*e^3*sgn(b*x + a) + 2*A*a*b^2*d*x^6*e^3*sgn(b*x +
a) + 18/5*B*a^2*b*d^2*x^5*e^2*sgn(b*x + a) + 18/5*A*a*b^2*d^2*x^5*e^2*sgn(b*x + a) + 3*B*a^2*b*d^3*x^4*e*sgn(b
*x + a) + 3*A*a*b^2*d^3*x^4*e*sgn(b*x + a) + B*a^2*b*d^4*x^3*sgn(b*x + a) + A*a*b^2*d^4*x^3*sgn(b*x + a) + 1/6
*B*a^3*x^6*e^4*sgn(b*x + a) + 1/2*A*a^2*b*x^6*e^4*sgn(b*x + a) + 4/5*B*a^3*d*x^5*e^3*sgn(b*x + a) + 12/5*A*a^2
*b*d*x^5*e^3*sgn(b*x + a) + 3/2*B*a^3*d^2*x^4*e^2*sgn(b*x + a) + 9/2*A*a^2*b*d^2*x^4*e^2*sgn(b*x + a) + 4/3*B*
a^3*d^3*x^3*e*sgn(b*x + a) + 4*A*a^2*b*d^3*x^3*e*sgn(b*x + a) + 1/2*B*a^3*d^4*x^2*sgn(b*x + a) + 3/2*A*a^2*b*d
^4*x^2*sgn(b*x + a) + 1/5*A*a^3*x^5*e^4*sgn(b*x + a) + A*a^3*d*x^4*e^3*sgn(b*x + a) + 2*A*a^3*d^2*x^3*e^2*sgn(
b*x + a) + 2*A*a^3*d^3*x^2*e*sgn(b*x + a) + A*a^3*d^4*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 552, normalized size = 1.85 \begin {gather*} \frac {\left (280 b^{3} B \,e^{4} x^{8}+315 x^{7} A \,b^{3} e^{4}+945 x^{7} B a \,b^{2} e^{4}+1260 x^{7} b^{3} B d \,e^{3}+1080 x^{6} A a \,b^{2} e^{4}+1440 x^{6} A \,b^{3} d \,e^{3}+1080 x^{6} B \,a^{2} b \,e^{4}+4320 x^{6} B a \,b^{2} d \,e^{3}+2160 x^{6} b^{3} B \,d^{2} e^{2}+1260 x^{5} A \,a^{2} b \,e^{4}+5040 x^{5} A a \,b^{2} d \,e^{3}+2520 x^{5} A \,b^{3} d^{2} e^{2}+420 x^{5} B \,a^{3} e^{4}+5040 x^{5} B \,a^{2} b d \,e^{3}+7560 x^{5} B a \,b^{2} d^{2} e^{2}+1680 x^{5} b^{3} B \,d^{3} e +504 x^{4} A \,a^{3} e^{4}+6048 x^{4} A \,a^{2} b d \,e^{3}+9072 x^{4} A a \,b^{2} d^{2} e^{2}+2016 x^{4} A \,b^{3} d^{3} e +2016 x^{4} B \,a^{3} d \,e^{3}+9072 x^{4} B \,a^{2} b \,d^{2} e^{2}+6048 x^{4} B a \,b^{2} d^{3} e +504 x^{4} b^{3} B \,d^{4}+2520 x^{3} A \,a^{3} d \,e^{3}+11340 x^{3} A \,a^{2} b \,d^{2} e^{2}+7560 x^{3} A a \,b^{2} d^{3} e +630 x^{3} A \,b^{3} d^{4}+3780 x^{3} B \,a^{3} d^{2} e^{2}+7560 x^{3} B \,a^{2} b \,d^{3} e +1890 x^{3} B a \,b^{2} d^{4}+5040 x^{2} A \,a^{3} d^{2} e^{2}+10080 x^{2} A \,a^{2} b \,d^{3} e +2520 x^{2} A a \,b^{2} d^{4}+3360 x^{2} B \,a^{3} d^{3} e +2520 x^{2} B \,a^{2} b \,d^{4}+5040 x A \,a^{3} d^{3} e +3780 x A \,a^{2} b \,d^{4}+1260 x B \,a^{3} d^{4}+2520 A \,a^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{2520 \left (b x +a \right )^{3}} \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)^(3/2),x)

[Out]

1/2520*x*(280*B*b^3*e^4*x^8+315*A*b^3*e^4*x^7+945*B*a*b^2*e^4*x^7+1260*B*b^3*d*e^3*x^7+1080*A*a*b^2*e^4*x^6+14
40*A*b^3*d*e^3*x^6+1080*B*a^2*b*e^4*x^6+4320*B*a*b^2*d*e^3*x^6+2160*B*b^3*d^2*e^2*x^6+1260*A*a^2*b*e^4*x^5+504
0*A*a*b^2*d*e^3*x^5+2520*A*b^3*d^2*e^2*x^5+420*B*a^3*e^4*x^5+5040*B*a^2*b*d*e^3*x^5+7560*B*a*b^2*d^2*e^2*x^5+1
680*B*b^3*d^3*e*x^5+504*A*a^3*e^4*x^4+6048*A*a^2*b*d*e^3*x^4+9072*A*a*b^2*d^2*e^2*x^4+2016*A*b^3*d^3*e*x^4+201
6*B*a^3*d*e^3*x^4+9072*B*a^2*b*d^2*e^2*x^4+6048*B*a*b^2*d^3*e*x^4+504*B*b^3*d^4*x^4+2520*A*a^3*d*e^3*x^3+11340
*A*a^2*b*d^2*e^2*x^3+7560*A*a*b^2*d^3*e*x^3+630*A*b^3*d^4*x^3+3780*B*a^3*d^2*e^2*x^3+7560*B*a^2*b*d^3*e*x^3+18
90*B*a*b^2*d^4*x^3+5040*A*a^3*d^2*e^2*x^2+10080*A*a^2*b*d^3*e*x^2+2520*A*a*b^2*d^4*x^2+3360*B*a^3*d^3*e*x^2+25
20*B*a^2*b*d^4*x^2+5040*A*a^3*d^3*e*x+3780*A*a^2*b*d^4*x+1260*B*a^3*d^4*x+2520*A*a^3*d^4)*((b*x+a)^2)^(3/2)/(b
*x+a)^3

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maxima [B]  time = 0.79, size = 1004, normalized size = 3.37

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)^(3/2),x, algorithm="maxima")

[Out]

1/9*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*e^4*x^4/b^2 - 13/72*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a*e^4*x^3/b^3 + 1/
4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*d^4*x - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^5*e^4*x/b^5 + 37/168*(b^2*
x^2 + 2*a*b*x + a^2)^(5/2)*B*a^2*e^4*x^2/b^4 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*a*d^4/b - 1/4*(b^2*x^2 +
2*a*b*x + a^2)^(3/2)*B*a^6*e^4/b^6 - 121/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^3*e^4*x/b^5 + 125/504*(b^2*x^
2 + 2*a*b*x + a^2)^(5/2)*B*a^4*e^4/b^6 + 1/8*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^3/b^2 + 1/4
*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4*x/b^4 - 1/2*(3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*
b*x + a^2)^(3/2)*a^3*x/b^3 + 1/2*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b^2 - 1/4*(B*
d^4 + 4*A*d^3*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b - 11/56*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(
5/2)*a*x^2/b^3 + 2/7*(3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^2/b^2 + 1/4*(4*B*d*e^3 + A*e^
4)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^5/b^5 - 1/2*(3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4
/b^4 + 1/2*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3/b^3 - 1/4*(B*d^4 + 4*A*d^3*e)*(b^2*x^
2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2 + 13/56*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^4 - 3/7*(
3*B*d^2*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*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)*x/b^2 - 69/280*(4*B*d*e^3 + A*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^5 + 17/35*(3*B*d^2
*e^2 + 2*A*d*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^4 - 7/15*(2*B*d^3*e + 3*A*d^2*e^2)*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a/b^3 + 1/5*(B*d^4 + 4*A*d^3*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)/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 )}^{3/2} \,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)^(3/2),x)

[Out]

int((A + B*x)*(d + e*x)^4*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), 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 {3}{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)**(3/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**4*((a + b*x)**2)**(3/2), x)

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