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

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

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Rubi [A]  time = 0.32, 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)^6 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac {e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^3 (A b-a B) (b d-a e)^3}{4 b^5}+\frac {B e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7}{8 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)^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.14, size = 320, normalized size = 1.24 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (14 a^3 \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 )+14 a^2 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 )+2 a 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 )+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 )\right )}{280 (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)^(3/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(14*a^3*(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)) + 14*a^2*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 + 45*
d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)) + 2*a*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)) + b^3*x^3*(2*A*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 2
0*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))))/(280*(a + b*x))

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IntegrateAlgebraic [F]  time = 3.91, 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 )^{3/2} \, 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)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.43, size = 325, normalized size = 1.25 \begin {gather*} \frac {1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac {1}{7} \, {\left (3 \, B b^{3} d e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{3} d^{2} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (A a^{3} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} + {\left (A a^{3} d e^{2} + {\left (B a^{2} b + A a b^{2}\right )} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{3} d^{2} e + {\left (B a^{3} + 3 \, A a^{2} 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)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.21, size = 594, normalized size = 2.29 \begin {gather*} \frac {1}{8} \, B b^{3} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, B b^{3} d x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B b^{3} d^{2} x^{6} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B b^{3} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{7} \, B a b^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, A b^{3} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, B a b^{2} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A b^{3} d x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{5} \, B a b^{2} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, A b^{3} d^{2} x^{5} e \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a b^{2} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{2} b x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, A a b^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{5} \, B a^{2} b d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{5} \, A a b^{2} d x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, B a^{2} b d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, A a b^{2} d^{2} x^{4} e \mathrm {sgn}\left (b x + a\right ) + B a^{2} b d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{5} \, B a^{3} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, A a^{2} b x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, B a^{3} d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, A a^{2} b d x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + B a^{3} d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a^{2} b d^{2} x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a^{3} d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{2} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A a^{3} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + A a^{3} d x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, A a^{3} d^{2} x^{2} e \mathrm {sgn}\left (b x + a\right ) + A a^{3} d^{3} 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)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 428, normalized size = 1.65 \begin {gather*} \frac {\left (35 B \,e^{3} b^{3} x^{7}+40 x^{6} A \,b^{3} e^{3}+120 x^{6} B \,e^{3} a \,b^{2}+120 x^{6} B \,b^{3} d \,e^{2}+140 x^{5} A a \,b^{2} e^{3}+140 x^{5} A \,b^{3} d \,e^{2}+140 x^{5} B \,e^{3} a^{2} b +420 x^{5} B a \,b^{2} d \,e^{2}+140 x^{5} B \,b^{3} d^{2} e +168 x^{4} A \,a^{2} b \,e^{3}+504 x^{4} A a \,b^{2} d \,e^{2}+168 x^{4} A \,b^{3} d^{2} e +56 x^{4} B \,a^{3} e^{3}+504 x^{4} B \,a^{2} b d \,e^{2}+504 x^{4} B a \,b^{2} d^{2} e +56 x^{4} B \,b^{3} d^{3}+70 x^{3} A \,a^{3} e^{3}+630 x^{3} A \,a^{2} b d \,e^{2}+630 x^{3} A a \,b^{2} d^{2} e +70 x^{3} A \,d^{3} b^{3}+210 x^{3} B \,a^{3} d \,e^{2}+630 x^{3} B \,a^{2} b \,d^{2} e +210 x^{3} B a \,b^{2} d^{3}+280 A \,a^{3} d \,e^{2} x^{2}+840 A \,a^{2} b \,d^{2} e \,x^{2}+280 A a \,b^{2} d^{3} x^{2}+280 B \,a^{3} d^{2} e \,x^{2}+280 B \,a^{2} b \,d^{3} x^{2}+420 x A \,a^{3} d^{2} e +420 x A \,d^{3} a^{2} b +140 x \,a^{3} B \,d^{3}+280 A \,d^{3} a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{280 \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)^3*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/280*x*(35*B*b^3*e^3*x^7+40*A*b^3*e^3*x^6+120*B*a*b^2*e^3*x^6+120*B*b^3*d*e^2*x^6+140*A*a*b^2*e^3*x^5+140*A*b
^3*d*e^2*x^5+140*B*a^2*b*e^3*x^5+420*B*a*b^2*d*e^2*x^5+140*B*b^3*d^2*e*x^5+168*A*a^2*b*e^3*x^4+504*A*a*b^2*d*e
^2*x^4+168*A*b^3*d^2*e*x^4+56*B*a^3*e^3*x^4+504*B*a^2*b*d*e^2*x^4+504*B*a*b^2*d^2*e*x^4+56*B*b^3*d^3*x^4+70*A*
a^3*e^3*x^3+630*A*a^2*b*d*e^2*x^3+630*A*a*b^2*d^2*e*x^3+70*A*b^3*d^3*x^3+210*B*a^3*d*e^2*x^3+630*B*a^2*b*d^2*e
*x^3+210*B*a*b^2*d^3*x^3+280*A*a^3*d*e^2*x^2+840*A*a^2*b*d^2*e*x^2+280*A*a*b^2*d^3*x^2+280*B*a^3*d^2*e*x^2+280
*B*a^2*b*d^3*x^2+420*A*a^3*d^2*e*x+420*A*a^2*b*d^3*x+140*B*a^3*d^3*x+280*A*a^3*d^3)*((b*x+a)^2)^(3/2)/(b*x+a)^
3

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maxima [B]  time = 0.56, size = 698, normalized size = 2.69 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B e^{3} x^{3}}{8 \, b^{2}} + \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{4} e^{3} x}{4 \, b^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a e^{3} x^{2}}{56 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a d^{3}}{4 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{5} e^{3}}{4 \, b^{5}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2} e^{3} x}{56 \, b^{4}} - \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{3} e^{3}}{280 \, 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 {3}{2}} a^{3} x}{4 \, 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 {3}{2}} a^{2} x}{4 \, 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 {3}{2}} a x}{4 \, 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 {5}{2}} x^{2}}{7 \, 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 {3}{2}} a^{4}}{4 \, b^{4}} + \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 {3}{2}} a^{3}}{4 \, 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 {3}{2}} a^{2}}{4 \, b^{2}} - \frac {3 \, {\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 x}{14 \, 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}} x}{2 \, b^{2}} + \frac {17 \, {\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^{2}}{70 \, b^{4}} - \frac {7 \, {\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}{10 \, 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}}}{5 \, 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)^(3/2),x, algorithm="maxima")

[Out]

1/8*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*e^3*x^3/b^2 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*d^3*x + 1/4*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*B*a^4*e^3*x/b^4 - 11/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a*e^3*x^2/b^3 + 1/4*(b^2*x^2
+ 2*a*b*x + a^2)^(3/2)*A*a*d^3/b + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*a^5*e^3/b^5 + 13/56*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*B*a^2*e^3*x/b^4 - 69/280*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*a^3*e^3/b^5 - 1/4*(3*B*d*e^2 + A*e^
3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3*x/b^3 + 3/4*(B*d^2*e + A*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2*x/b
^2 - 1/4*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a*x/b + 1/7*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*
x + a^2)^(5/2)*x^2/b^2 - 1/4*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^4/b^4 + 3/4*(B*d^2*e + A*d*
e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^3/b^3 - 1/4*(B*d^3 + 3*A*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*a^2/b^2
- 3/14*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*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)*x/b^2 + 17/70*(3*B*d*e^2 + A*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^4 - 7/10*(B*d^2*e + A*
d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a/b^3 + 1/5*(B*d^3 + 3*A*d^2*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 )}^3\,{\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)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

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

[Out]

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

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