3.16.21 \(\int (A+B x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=259 \[ \frac {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (-4 a B e+A b e+3 b B d)}{9 b^5}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{8 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{7 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^3}{6 b^5}+\frac {B e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \]

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Rubi [A]  time = 0.46, antiderivative size = 259, 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 {e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (-4 a B e+A b e+3 b B d)}{9 b^5}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{8 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{7 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^3}{6 b^5}+\frac {B e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.20, size = 478, normalized size = 1.85 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (126 a^5 \left (5 A \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+B x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )+210 a^4 b x \left (3 A \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )+60 a^3 b^2 x^2 \left (7 A \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )+90 a^2 b^3 x^3 \left (2 A \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )\right )+5 a b^4 x^4 \left (9 A \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+5 B x \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )+b^5 x^5 \left (5 A \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+3 B x \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right )\right )}{2520 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(126*a^5*(5*A*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + B*x*(10*d^3 + 20*d^2*e*x + 15
*d*e^2*x^2 + 4*e^3*x^3)) + 210*a^4*b*x*(3*A*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + B*x*(20*d^3 + 4
5*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)) + 60*a^3*b^2*x^2*(7*A*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3
) + 3*B*x*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)) + 90*a^2*b^3*x^3*(2*A*(35*d^3 + 84*d^2*e*x + 70*d
*e^2*x^2 + 20*e^3*x^3) + B*x*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3)) + 5*a*b^4*x^4*(9*A*(56*d^3 +
140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 5*B*x*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)) + b^5*
x^5*(5*A*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 3*B*x*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 +
84*e^3*x^3))))/(2520*(a + b*x))

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IntegrateAlgebraic [F]  time = 7.16, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^3 \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)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

Defer[IntegrateAlgebraic][(A + B*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2), x]

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fricas [B]  time = 0.44, size = 532, normalized size = 2.05 \begin {gather*} \frac {1}{10} \, B b^{5} e^{3} x^{10} + A a^{5} d^{3} x + \frac {1}{9} \, {\left (3 \, B b^{5} d e^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (3 \, B b^{5} d^{2} e + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{2} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{3} + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{2} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} + 15 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} + 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e + 15 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{2} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{5} e^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} + 15 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{5} d e^{2} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{5} d^{2} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/10*B*b^5*e^3*x^10 + A*a^5*d^3*x + 1/9*(3*B*b^5*d*e^2 + (5*B*a*b^4 + A*b^5)*e^3)*x^9 + 1/8*(3*B*b^5*d^2*e + 3
*(5*B*a*b^4 + A*b^5)*d*e^2 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^3)*x^8 + 1/7*(B*b^5*d^3 + 3*(5*B*a*b^4 + A*b^5)*d^2*e
+ 15*(2*B*a^2*b^3 + A*a*b^4)*d*e^2 + 10*(B*a^3*b^2 + A*a^2*b^3)*e^3)*x^7 + 1/6*((5*B*a*b^4 + A*b^5)*d^3 + 15*
(2*B*a^2*b^3 + A*a*b^4)*d^2*e + 30*(B*a^3*b^2 + A*a^2*b^3)*d*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*e^3)*x^6 + 1/5*(5
*(2*B*a^2*b^3 + A*a*b^4)*d^3 + 30*(B*a^3*b^2 + A*a^2*b^3)*d^2*e + 15*(B*a^4*b + 2*A*a^3*b^2)*d*e^2 + (B*a^5 +
5*A*a^4*b)*e^3)*x^5 + 1/4*(A*a^5*e^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*d^3 + 15*(B*a^4*b + 2*A*a^3*b^2)*d^2*e + 3*(
B*a^5 + 5*A*a^4*b)*d*e^2)*x^4 + 1/3*(3*A*a^5*d*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^3 + 3*(B*a^5 + 5*A*a^4*b)*d^2
*e)*x^3 + 1/2*(3*A*a^5*d^2*e + (B*a^5 + 5*A*a^4*b)*d^3)*x^2

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giac [B]  time = 0.22, size = 934, normalized size = 3.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/10*B*b^5*x^10*e^3*sgn(b*x + a) + 1/3*B*b^5*d*x^9*e^2*sgn(b*x + a) + 3/8*B*b^5*d^2*x^8*e*sgn(b*x + a) + 1/7*B
*b^5*d^3*x^7*sgn(b*x + a) + 5/9*B*a*b^4*x^9*e^3*sgn(b*x + a) + 1/9*A*b^5*x^9*e^3*sgn(b*x + a) + 15/8*B*a*b^4*d
*x^8*e^2*sgn(b*x + a) + 3/8*A*b^5*d*x^8*e^2*sgn(b*x + a) + 15/7*B*a*b^4*d^2*x^7*e*sgn(b*x + a) + 3/7*A*b^5*d^2
*x^7*e*sgn(b*x + a) + 5/6*B*a*b^4*d^3*x^6*sgn(b*x + a) + 1/6*A*b^5*d^3*x^6*sgn(b*x + a) + 5/4*B*a^2*b^3*x^8*e^
3*sgn(b*x + a) + 5/8*A*a*b^4*x^8*e^3*sgn(b*x + a) + 30/7*B*a^2*b^3*d*x^7*e^2*sgn(b*x + a) + 15/7*A*a*b^4*d*x^7
*e^2*sgn(b*x + a) + 5*B*a^2*b^3*d^2*x^6*e*sgn(b*x + a) + 5/2*A*a*b^4*d^2*x^6*e*sgn(b*x + a) + 2*B*a^2*b^3*d^3*
x^5*sgn(b*x + a) + A*a*b^4*d^3*x^5*sgn(b*x + a) + 10/7*B*a^3*b^2*x^7*e^3*sgn(b*x + a) + 10/7*A*a^2*b^3*x^7*e^3
*sgn(b*x + a) + 5*B*a^3*b^2*d*x^6*e^2*sgn(b*x + a) + 5*A*a^2*b^3*d*x^6*e^2*sgn(b*x + a) + 6*B*a^3*b^2*d^2*x^5*
e*sgn(b*x + a) + 6*A*a^2*b^3*d^2*x^5*e*sgn(b*x + a) + 5/2*B*a^3*b^2*d^3*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d^3*x
^4*sgn(b*x + a) + 5/6*B*a^4*b*x^6*e^3*sgn(b*x + a) + 5/3*A*a^3*b^2*x^6*e^3*sgn(b*x + a) + 3*B*a^4*b*d*x^5*e^2*
sgn(b*x + a) + 6*A*a^3*b^2*d*x^5*e^2*sgn(b*x + a) + 15/4*B*a^4*b*d^2*x^4*e*sgn(b*x + a) + 15/2*A*a^3*b^2*d^2*x
^4*e*sgn(b*x + a) + 5/3*B*a^4*b*d^3*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^3*x^3*sgn(b*x + a) + 1/5*B*a^5*x^5*e^3
*sgn(b*x + a) + A*a^4*b*x^5*e^3*sgn(b*x + a) + 3/4*B*a^5*d*x^4*e^2*sgn(b*x + a) + 15/4*A*a^4*b*d*x^4*e^2*sgn(b
*x + a) + B*a^5*d^2*x^3*e*sgn(b*x + a) + 5*A*a^4*b*d^2*x^3*e*sgn(b*x + a) + 1/2*B*a^5*d^3*x^2*sgn(b*x + a) + 5
/2*A*a^4*b*d^3*x^2*sgn(b*x + a) + 1/4*A*a^5*x^4*e^3*sgn(b*x + a) + A*a^5*d*x^3*e^2*sgn(b*x + a) + 3/2*A*a^5*d^
2*x^2*e*sgn(b*x + a) + A*a^5*d^3*x*sgn(b*x + a)

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maple [B]  time = 0.05, size = 676, normalized size = 2.61 \begin {gather*} \frac {\left (252 B \,e^{3} b^{5} x^{9}+280 x^{8} A \,b^{5} e^{3}+1400 x^{8} B \,e^{3} a \,b^{4}+840 x^{8} B \,b^{5} d \,e^{2}+1575 x^{7} A a \,b^{4} e^{3}+945 x^{7} A \,b^{5} d \,e^{2}+3150 x^{7} B \,e^{3} a^{2} b^{3}+4725 x^{7} B a \,b^{4} d \,e^{2}+945 x^{7} B \,b^{5} d^{2} e +3600 x^{6} A \,a^{2} b^{3} e^{3}+5400 x^{6} A a \,b^{4} d \,e^{2}+1080 x^{6} A \,b^{5} d^{2} e +3600 x^{6} B \,e^{3} a^{3} b^{2}+10800 x^{6} B \,a^{2} b^{3} d \,e^{2}+5400 x^{6} B a \,b^{4} d^{2} e +360 x^{6} B \,b^{5} d^{3}+4200 x^{5} A \,a^{3} b^{2} e^{3}+12600 x^{5} A \,a^{2} b^{3} d \,e^{2}+6300 x^{5} A a \,b^{4} d^{2} e +420 x^{5} A \,d^{3} b^{5}+2100 x^{5} B \,e^{3} a^{4} b +12600 x^{5} B \,a^{3} b^{2} d \,e^{2}+12600 x^{5} B \,a^{2} b^{3} d^{2} e +2100 x^{5} B a \,b^{4} d^{3}+2520 x^{4} A \,a^{4} b \,e^{3}+15120 x^{4} A \,a^{3} b^{2} d \,e^{2}+15120 x^{4} A \,a^{2} b^{3} d^{2} e +2520 x^{4} A \,d^{3} a \,b^{4}+504 x^{4} B \,e^{3} a^{5}+7560 x^{4} B \,a^{4} b d \,e^{2}+15120 x^{4} B \,a^{3} b^{2} d^{2} e +5040 x^{4} B \,a^{2} b^{3} d^{3}+630 x^{3} A \,a^{5} e^{3}+9450 x^{3} A \,a^{4} b d \,e^{2}+18900 x^{3} A \,a^{3} b^{2} d^{2} e +6300 x^{3} A \,d^{3} a^{2} b^{3}+1890 x^{3} B \,a^{5} d \,e^{2}+9450 x^{3} B \,a^{4} b \,d^{2} e +6300 x^{3} B \,a^{3} b^{2} d^{3}+2520 x^{2} A \,a^{5} d \,e^{2}+12600 x^{2} A \,a^{4} b \,d^{2} e +8400 x^{2} A \,d^{3} a^{3} b^{2}+2520 x^{2} B \,a^{5} d^{2} e +4200 x^{2} B \,a^{4} b \,d^{3}+3780 x A \,a^{5} d^{2} e +6300 x A \,d^{3} a^{4} b +1260 x B \,a^{5} d^{3}+2520 A \,d^{3} a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{2520 \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)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/2520*x*(252*B*b^5*e^3*x^9+280*A*b^5*e^3*x^8+1400*B*a*b^4*e^3*x^8+840*B*b^5*d*e^2*x^8+1575*A*a*b^4*e^3*x^7+94
5*A*b^5*d*e^2*x^7+3150*B*a^2*b^3*e^3*x^7+4725*B*a*b^4*d*e^2*x^7+945*B*b^5*d^2*e*x^7+3600*A*a^2*b^3*e^3*x^6+540
0*A*a*b^4*d*e^2*x^6+1080*A*b^5*d^2*e*x^6+3600*B*a^3*b^2*e^3*x^6+10800*B*a^2*b^3*d*e^2*x^6+5400*B*a*b^4*d^2*e*x
^6+360*B*b^5*d^3*x^6+4200*A*a^3*b^2*e^3*x^5+12600*A*a^2*b^3*d*e^2*x^5+6300*A*a*b^4*d^2*e*x^5+420*A*b^5*d^3*x^5
+2100*B*a^4*b*e^3*x^5+12600*B*a^3*b^2*d*e^2*x^5+12600*B*a^2*b^3*d^2*e*x^5+2100*B*a*b^4*d^3*x^5+2520*A*a^4*b*e^
3*x^4+15120*A*a^3*b^2*d*e^2*x^4+15120*A*a^2*b^3*d^2*e*x^4+2520*A*a*b^4*d^3*x^4+504*B*a^5*e^3*x^4+7560*B*a^4*b*
d*e^2*x^4+15120*B*a^3*b^2*d^2*e*x^4+5040*B*a^2*b^3*d^3*x^4+630*A*a^5*e^3*x^3+9450*A*a^4*b*d*e^2*x^3+18900*A*a^
3*b^2*d^2*e*x^3+6300*A*a^2*b^3*d^3*x^3+1890*B*a^5*d*e^2*x^3+9450*B*a^4*b*d^2*e*x^3+6300*B*a^3*b^2*d^3*x^3+2520
*A*a^5*d*e^2*x^2+12600*A*a^4*b*d^2*e*x^2+8400*A*a^3*b^2*d^3*x^2+2520*B*a^5*d^2*e*x^2+4200*B*a^4*b*d^3*x^2+3780
*A*a^5*d^2*e*x+6300*A*a^4*b*d^3*x+1260*B*a^5*d^3*x+2520*A*a^5*d^3)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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maxima [B]  time = 0.54, size = 698, normalized size = 2.69 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B e^{3} x^{3}}{10 \, b^{2}} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{4} e^{3} x}{6 \, b^{4}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a e^{3} x^{2}}{90 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a d^{3}}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{5} e^{3}}{6 \, b^{5}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{2} e^{3} x}{180 \, b^{4}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B a^{3} e^{3}}{1260 \, b^{5}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x}{6 \, b^{3}} + \frac {{\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{6 \, b} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, b^{2}} - \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4}}{6 \, b^{4}} + \frac {{\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{6 \, b^{2}} - \frac {11 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x}{72 \, b^{3}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x}{8 \, b^{2}} + \frac {83 \, {\left (3 \, B d e^{2} + A e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2}}{504 \, b^{4}} - \frac {27 \, {\left (B d^{2} e + A d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a}{56 \, b^{3}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/10*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*e^3*x^3/b^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*d^3*x + 1/6*(b^2*x^
2 + 2*a*b*x + a^2)^(5/2)*B*a^4*e^3*x/b^4 - 13/90*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a*e^3*x^2/b^3 + 1/6*(b^2*x^
2 + 2*a*b*x + a^2)^(5/2)*A*a*d^3/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^5*e^3/b^5 + 29/180*(b^2*x^2 + 2*a
*b*x + a^2)^(7/2)*B*a^2*e^3*x/b^4 - 209/1260*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*a^3*e^3/b^5 - 1/6*(3*B*d*e^2 +
A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 1/2*(B*d^2*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2
*x/b^2 - 1/6*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*x/b + 1/9*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*
a*b*x + a^2)^(7/2)*x^2/b^2 - 1/6*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 1/2*(B*d^2*e +
A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2
/b^2 - 11/72*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 3/8*(B*d^2*e + A*d*e^2)*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*x/b^2 + 83/504*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 27/56*(B*d^2
*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(B*d^3 + 3*A*d^2*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 )}^3\,{\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)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((A + B*x)*(d + e*x)^3*(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 )^{3} \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)**3*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**3*((a + b*x)**2)**(5/2), x)

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