3.1286 \(\int (A+B x) (d+e x)^3 (a+c x^2) \, dx\)

Optimal. Leaf size=108 \[ \frac{(d+e x)^5 \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac{(d+e x)^4 \left (a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac{c (d+e x)^6 (3 B d-A e)}{6 e^4}+\frac{B c (d+e x)^7}{7 e^4} \]

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^4)/(4*e^4) + ((3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^5)/(5*e^4) -
(c*(3*B*d - A*e)*(d + e*x)^6)/(6*e^4) + (B*c*(d + e*x)^7)/(7*e^4)

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Rubi [A]  time = 0.114393, 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} \[ \frac{(d+e x)^5 \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac{(d+e x)^4 \left (a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac{c (d+e x)^6 (3 B d-A e)}{6 e^4}+\frac{B c (d+e x)^7}{7 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.041079, size = 150, normalized size = 1.39 \[ \frac{1}{5} e x^5 \left (a B e^2+3 A c d e+3 B c d^2\right )+\frac{1}{4} x^4 \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+\frac{1}{3} d x^3 \left (3 a A e^2+3 a B d e+A c d^2\right )+\frac{1}{2} a d^2 x^2 (3 A e+B d)+a A d^3 x+\frac{1}{6} c e^2 x^6 (A e+3 B d)+\frac{1}{7} B c e^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*A*d^3*x + (a*d^2*(B*d + 3*A*e)*x^2)/2 + (d*(A*c*d^2 + 3*a*B*d*e + 3*a*A*e^2)*x^3)/3 + ((B*c*d^3 + 3*A*c*d^2*
e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/4 + (e*(3*B*c*d^2 + 3*A*c*d*e + a*B*e^2)*x^5)/5 + (c*e^2*(3*B*d + A*e)*x^6)/6
+ (B*c*e^3*x^7)/7

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Maple [A]  time = 0., size = 151, normalized size = 1.4 \begin{align*}{\frac{B{e}^{3}c{x}^{7}}{7}}+{\frac{ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) c{x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) c+B{e}^{3}a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) c+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{3}c+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) a{x}^{2}}{2}}+A{d}^{3}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3*(c*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.08348, size = 200, normalized size = 1.85 \begin{align*} \frac{1}{7} \, B c e^{3} x^{7} + \frac{1}{6} \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{6} + A a d^{3} x + \frac{1}{5} \,{\left (3 \, B c d^{2} e + 3 \, A c d e^{2} + B a e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{3} + 3 \, A c d^{2} e + 3 \, B a d e^{2} + A a e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (A c d^{3} + 3 \, B a d^{2} e + 3 \, A a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59941, size = 390, normalized size = 3.61 \begin{align*} \frac{1}{7} x^{7} e^{3} c B + \frac{1}{2} x^{6} e^{2} d c B + \frac{1}{6} x^{6} e^{3} c A + \frac{3}{5} x^{5} e d^{2} c B + \frac{1}{5} x^{5} e^{3} a B + \frac{3}{5} x^{5} e^{2} d c A + \frac{1}{4} x^{4} d^{3} c B + \frac{3}{4} x^{4} e^{2} d a B + \frac{3}{4} x^{4} e d^{2} c A + \frac{1}{4} x^{4} e^{3} a A + x^{3} e d^{2} a B + \frac{1}{3} x^{3} d^{3} c A + x^{3} e^{2} d a A + \frac{1}{2} x^{2} d^{3} a B + \frac{3}{2} x^{2} e d^{2} a A + x d^{3} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^3*c*B + 1/2*x^6*e^2*d*c*B + 1/6*x^6*e^3*c*A + 3/5*x^5*e*d^2*c*B + 1/5*x^5*e^3*a*B + 3/5*x^5*e^2*d*c*
A + 1/4*x^4*d^3*c*B + 3/4*x^4*e^2*d*a*B + 3/4*x^4*e*d^2*c*A + 1/4*x^4*e^3*a*A + x^3*e*d^2*a*B + 1/3*x^3*d^3*c*
A + x^3*e^2*d*a*A + 1/2*x^2*d^3*a*B + 3/2*x^2*e*d^2*a*A + x*d^3*a*A

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Sympy [A]  time = 0.133515, size = 173, normalized size = 1.6 \begin{align*} A a d^{3} x + \frac{B c e^{3} x^{7}}{7} + x^{6} \left (\frac{A c e^{3}}{6} + \frac{B c d e^{2}}{2}\right ) + x^{5} \left (\frac{3 A c d e^{2}}{5} + \frac{B a e^{3}}{5} + \frac{3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac{A a e^{3}}{4} + \frac{3 A c d^{2} e}{4} + \frac{3 B a d e^{2}}{4} + \frac{B c d^{3}}{4}\right ) + x^{3} \left (A a d e^{2} + \frac{A c d^{3}}{3} + B a d^{2} e\right ) + x^{2} \left (\frac{3 A a d^{2} e}{2} + \frac{B a d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3*(c*x**2+a),x)

[Out]

A*a*d**3*x + B*c*e**3*x**7/7 + x**6*(A*c*e**3/6 + B*c*d*e**2/2) + x**5*(3*A*c*d*e**2/5 + B*a*e**3/5 + 3*B*c*d*
*2*e/5) + x**4*(A*a*e**3/4 + 3*A*c*d**2*e/4 + 3*B*a*d*e**2/4 + B*c*d**3/4) + x**3*(A*a*d*e**2 + A*c*d**3/3 + B
*a*d**2*e) + x**2*(3*A*a*d**2*e/2 + B*a*d**3/2)

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Giac [A]  time = 1.28493, size = 216, normalized size = 2. \begin{align*} \frac{1}{7} \, B c x^{7} e^{3} + \frac{1}{2} \, B c d x^{6} e^{2} + \frac{3}{5} \, B c d^{2} x^{5} e + \frac{1}{4} \, B c d^{3} x^{4} + \frac{1}{6} \, A c x^{6} e^{3} + \frac{3}{5} \, A c d x^{5} e^{2} + \frac{3}{4} \, A c d^{2} x^{4} e + \frac{1}{3} \, A c d^{3} x^{3} + \frac{1}{5} \, B a x^{5} e^{3} + \frac{3}{4} \, B a d x^{4} e^{2} + B a d^{2} x^{3} e + \frac{1}{2} \, B a d^{3} x^{2} + \frac{1}{4} \, A a x^{4} e^{3} + A a d x^{3} e^{2} + \frac{3}{2} \, A a d^{2} x^{2} e + A a d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*B*c*x^7*e^3 + 1/2*B*c*d*x^6*e^2 + 3/5*B*c*d^2*x^5*e + 1/4*B*c*d^3*x^4 + 1/6*A*c*x^6*e^3 + 3/5*A*c*d*x^5*e^
2 + 3/4*A*c*d^2*x^4*e + 1/3*A*c*d^3*x^3 + 1/5*B*a*x^5*e^3 + 3/4*B*a*d*x^4*e^2 + B*a*d^2*x^3*e + 1/2*B*a*d^3*x^
2 + 1/4*A*a*x^4*e^3 + A*a*d*x^3*e^2 + 3/2*A*a*d^2*x^2*e + A*a*d^3*x