3.1287 \(\int (A+B x) (d+e x)^2 (a+c x^2) \, dx\)

Optimal. Leaf size=108 \[ \frac{(d+e x)^4 \left (a B e^2-2 A c d e+3 B c d^2\right )}{4 e^4}-\frac{(d+e x)^3 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4}-\frac{c (d+e x)^5 (3 B d-A e)}{5 e^4}+\frac{B c (d+e x)^6}{6 e^4} \]

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^3)/(3*e^4) + ((3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^4)/(4*e^4) -
(c*(3*B*d - A*e)*(d + e*x)^5)/(5*e^4) + (B*c*(d + e*x)^6)/(6*e^4)

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Rubi [A]  time = 0.0851988, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {772} \[ \frac{(d+e x)^4 \left (a B e^2-2 A c d e+3 B c d^2\right )}{4 e^4}-\frac{(d+e x)^3 \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4}-\frac{c (d+e x)^5 (3 B d-A e)}{5 e^4}+\frac{B c (d+e x)^6}{6 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^2*(a + c*x^2),x]

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^3)/(3*e^4) + ((3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^4)/(4*e^4) -
(c*(3*B*d - A*e)*(d + e*x)^5)/(5*e^4) + (B*c*(d + e*x)^6)/(6*e^4)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^2 \left (a+c x^2\right ) \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right ) (d+e x)^2}{e^3}+\frac{\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^3}{e^3}+\frac{c (-3 B d+A e) (d+e x)^4}{e^3}+\frac{B c (d+e x)^5}{e^3}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^3}{3 e^4}+\frac{\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^4}{4 e^4}-\frac{c (3 B d-A e) (d+e x)^5}{5 e^4}+\frac{B c (d+e x)^6}{6 e^4}\\ \end{align*}

Mathematica [A]  time = 0.0288112, size = 106, normalized size = 0.98 \[ \frac{1}{4} x^4 \left (a B e^2+2 A c d e+B c d^2\right )+\frac{1}{3} x^3 \left (a A e^2+2 a B d e+A c d^2\right )+\frac{1}{2} a d x^2 (2 A e+B d)+a A d^2 x+\frac{1}{5} c e x^5 (A e+2 B d)+\frac{1}{6} B c e^2 x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^2*(a + c*x^2),x]

[Out]

a*A*d^2*x + (a*d*(B*d + 2*A*e)*x^2)/2 + ((A*c*d^2 + 2*a*B*d*e + a*A*e^2)*x^3)/3 + ((B*c*d^2 + 2*A*c*d*e + a*B*
e^2)*x^4)/4 + (c*e*(2*B*d + A*e)*x^5)/5 + (B*c*e^2*x^6)/6

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Maple [A]  time = 0., size = 103, normalized size = 1. \begin{align*}{\frac{Bc{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( A{e}^{2}+2\,Bde \right ) c{x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ) c+aB{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( Ac{d}^{2}+ \left ( A{e}^{2}+2\,Bde \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Ade+B{d}^{2} \right ) a{x}^{2}}{2}}+A{d}^{2}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^2*(c*x^2+a),x)

[Out]

1/6*B*c*e^2*x^6+1/5*(A*e^2+2*B*d*e)*c*x^5+1/4*((2*A*d*e+B*d^2)*c+a*B*e^2)*x^4+1/3*(A*c*d^2+(A*e^2+2*B*d*e)*a)*
x^3+1/2*(2*A*d*e+B*d^2)*a*x^2+A*d^2*a*x

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Maxima [A]  time = 1.16354, size = 138, normalized size = 1.28 \begin{align*} \frac{1}{6} \, B c e^{2} x^{6} + \frac{1}{5} \,{\left (2 \, B c d e + A c e^{2}\right )} x^{5} + A a d^{2} x + \frac{1}{4} \,{\left (B c d^{2} + 2 \, A c d e + B a e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A c d^{2} + 2 \, B a d e + A a e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a d^{2} + 2 \, A a d e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(c*x^2+a),x, algorithm="maxima")

[Out]

1/6*B*c*e^2*x^6 + 1/5*(2*B*c*d*e + A*c*e^2)*x^5 + A*a*d^2*x + 1/4*(B*c*d^2 + 2*A*c*d*e + B*a*e^2)*x^4 + 1/3*(A
*c*d^2 + 2*B*a*d*e + A*a*e^2)*x^3 + 1/2*(B*a*d^2 + 2*A*a*d*e)*x^2

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Fricas [A]  time = 1.61803, size = 277, normalized size = 2.56 \begin{align*} \frac{1}{6} x^{6} e^{2} c B + \frac{2}{5} x^{5} e d c B + \frac{1}{5} x^{5} e^{2} c A + \frac{1}{4} x^{4} d^{2} c B + \frac{1}{4} x^{4} e^{2} a B + \frac{1}{2} x^{4} e d c A + \frac{2}{3} x^{3} e d a B + \frac{1}{3} x^{3} d^{2} c A + \frac{1}{3} x^{3} e^{2} a A + \frac{1}{2} x^{2} d^{2} a B + x^{2} e d a A + x d^{2} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(c*x^2+a),x, algorithm="fricas")

[Out]

1/6*x^6*e^2*c*B + 2/5*x^5*e*d*c*B + 1/5*x^5*e^2*c*A + 1/4*x^4*d^2*c*B + 1/4*x^4*e^2*a*B + 1/2*x^4*e*d*c*A + 2/
3*x^3*e*d*a*B + 1/3*x^3*d^2*c*A + 1/3*x^3*e^2*a*A + 1/2*x^2*d^2*a*B + x^2*e*d*a*A + x*d^2*a*A

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Sympy [A]  time = 0.135153, size = 119, normalized size = 1.1 \begin{align*} A a d^{2} x + \frac{B c e^{2} x^{6}}{6} + x^{5} \left (\frac{A c e^{2}}{5} + \frac{2 B c d e}{5}\right ) + x^{4} \left (\frac{A c d e}{2} + \frac{B a e^{2}}{4} + \frac{B c d^{2}}{4}\right ) + x^{3} \left (\frac{A a e^{2}}{3} + \frac{A c d^{2}}{3} + \frac{2 B a d e}{3}\right ) + x^{2} \left (A a d e + \frac{B a d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**2*(c*x**2+a),x)

[Out]

A*a*d**2*x + B*c*e**2*x**6/6 + x**5*(A*c*e**2/5 + 2*B*c*d*e/5) + x**4*(A*c*d*e/2 + B*a*e**2/4 + B*c*d**2/4) +
x**3*(A*a*e**2/3 + A*c*d**2/3 + 2*B*a*d*e/3) + x**2*(A*a*d*e + B*a*d**2/2)

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Giac [A]  time = 1.17833, size = 153, normalized size = 1.42 \begin{align*} \frac{1}{6} \, B c x^{6} e^{2} + \frac{2}{5} \, B c d x^{5} e + \frac{1}{4} \, B c d^{2} x^{4} + \frac{1}{5} \, A c x^{5} e^{2} + \frac{1}{2} \, A c d x^{4} e + \frac{1}{3} \, A c d^{2} x^{3} + \frac{1}{4} \, B a x^{4} e^{2} + \frac{2}{3} \, B a d x^{3} e + \frac{1}{2} \, B a d^{2} x^{2} + \frac{1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2*(c*x^2+a),x, algorithm="giac")

[Out]

1/6*B*c*x^6*e^2 + 2/5*B*c*d*x^5*e + 1/4*B*c*d^2*x^4 + 1/5*A*c*x^5*e^2 + 1/2*A*c*d*x^4*e + 1/3*A*c*d^2*x^3 + 1/
4*B*a*x^4*e^2 + 2/3*B*a*d*x^3*e + 1/2*B*a*d^2*x^2 + 1/3*A*a*x^3*e^2 + A*a*d*x^2*e + A*a*d^2*x