3.1296 \(\int \frac{(A+B x) (a+c x^2)}{(d+e x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0679083, 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} \[ -\frac{a B e^2-2 A c d e+3 B c d^2}{5 e^4 (d+e x)^5}+\frac{\left (a e^2+c d^2\right ) (B d-A e)}{6 e^4 (d+e x)^6}+\frac{c (3 B d-A e)}{4 e^4 (d+e x)^4}-\frac{B c}{3 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/(d + e*x)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0465528, size = 87, normalized size = 0.81 \[ -\frac{10 a A e^3+2 a B e^2 (d+6 e x)+A c e \left (d^2+6 d e x+15 e^2 x^2\right )+B c \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^7,x]

[Out]

-(10*a*A*e^3 + 2*a*B*e^2*(d + 6*e*x) + A*c*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + B*c*(d^3 + 6*d^2*e*x + 15*d*e^2*x^
2 + 20*e^3*x^3))/(60*e^4*(d + e*x)^6)

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Maple [A]  time = 0.006, size = 110, normalized size = 1. \begin{align*} -{\frac{Bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( Ae-3\,Bd \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{-2\,Acde+aB{e}^{2}+3\,Bc{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{aA{e}^{3}+Ac{d}^{2}e-aBd{e}^{2}-Bc{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)/(e*x+d)^7,x)

[Out]

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

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Maxima [A]  time = 1.0513, size = 207, normalized size = 1.92 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + A c d^{2} e + 2 \, B a d e^{2} + 10 \, A a e^{3} + 15 \,{\left (B c d e^{2} + A c e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e + A c d e^{2} + 2 \, B a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + A*c*d^2*e + 2*B*a*d*e^2 + 10*A*a*e^3 + 15*(B*c*d*e^2 + A*c*e^3)*x^2 + 6*(B*c
*d^2*e + A*c*d*e^2 + 2*B*a*e^3)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2
+ 6*d^5*e^5*x + d^6*e^4)

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Fricas [A]  time = 1.78554, size = 332, normalized size = 3.07 \begin{align*} -\frac{20 \, B c e^{3} x^{3} + B c d^{3} + A c d^{2} e + 2 \, B a d e^{2} + 10 \, A a e^{3} + 15 \,{\left (B c d e^{2} + A c e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e + A c d e^{2} + 2 \, B a e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + A*c*d^2*e + 2*B*a*d*e^2 + 10*A*a*e^3 + 15*(B*c*d*e^2 + A*c*e^3)*x^2 + 6*(B*c
*d^2*e + A*c*d*e^2 + 2*B*a*e^3)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2
+ 6*d^5*e^5*x + d^6*e^4)

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Sympy [A]  time = 35.657, size = 172, normalized size = 1.59 \begin{align*} - \frac{10 A a e^{3} + A c d^{2} e + 2 B a d e^{2} + B c d^{3} + 20 B c e^{3} x^{3} + x^{2} \left (15 A c e^{3} + 15 B c d e^{2}\right ) + x \left (6 A c d e^{2} + 12 B a e^{3} + 6 B c d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)/(e*x+d)**7,x)

[Out]

-(10*A*a*e**3 + A*c*d**2*e + 2*B*a*d*e**2 + B*c*d**3 + 20*B*c*e**3*x**3 + x**2*(15*A*c*e**3 + 15*B*c*d*e**2) +
x*(6*A*c*d*e**2 + 12*B*a*e**3 + 6*B*c*d**2*e))/(60*d**6*e**4 + 360*d**5*e**5*x + 900*d**4*e**6*x**2 + 1200*d*
*3*e**7*x**3 + 900*d**2*e**8*x**4 + 360*d*e**9*x**5 + 60*e**10*x**6)

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Giac [A]  time = 1.24666, size = 126, normalized size = 1.17 \begin{align*} -\frac{{\left (20 \, B c x^{3} e^{3} + 15 \, B c d x^{2} e^{2} + 6 \, B c d^{2} x e + B c d^{3} + 15 \, A c x^{2} e^{3} + 6 \, A c d x e^{2} + A c d^{2} e + 12 \, B a x e^{3} + 2 \, B a d e^{2} + 10 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(20*B*c*x^3*e^3 + 15*B*c*d*x^2*e^2 + 6*B*c*d^2*x*e + B*c*d^3 + 15*A*c*x^2*e^3 + 6*A*c*d*x*e^2 + A*c*d^2*
e + 12*B*a*x*e^3 + 2*B*a*d*e^2 + 10*A*a*e^3)*e^(-4)/(x*e + d)^6