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

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

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Rubi [A]  time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {772} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 87, normalized size = 0.81 \begin {gather*} -\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 (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(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))/(e^4*(d + e*x)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 153, normalized size = 1.42 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.17, size = 93, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.05, size = 110, normalized size = 1.02 \begin {gather*} -\frac {B c}{3 \left (e x +d \right )^{3} e^{4}}-\frac {\left (A e -3 B d \right ) c}{4 \left (e x +d \right )^{4} e^{4}}-\frac {a A \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}}{6 \left (e x +d \right )^{6} e^{4}}-\frac {-2 A c d e +B a \,e^{2}+3 B c \,d^{2}}{5 \left (e x +d \right )^{5} e^{4}} \end {gather*}

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]

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

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maxima [A]  time = 0.59, size = 153, normalized size = 1.42 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.78, size = 150, normalized size = 1.39 \begin {gather*} -\frac {\frac {B\,c\,d^3+A\,c\,d^2\,e+2\,B\,a\,d\,e^2+10\,A\,a\,e^3}{60\,e^4}+\frac {x\,\left (B\,c\,d^2+A\,c\,d\,e+2\,B\,a\,e^2\right )}{10\,e^3}+\frac {B\,c\,x^3}{3\,e}+\frac {c\,x^2\,\left (A\,e+B\,d\right )}{4\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 13.57, size = 173, normalized size = 1.60 \begin {gather*} \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 {gather*}

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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