3.1295 \(\int \frac {(A+B x) (a+c x^2)}{(d+e x)^6} \, dx\)

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

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/(d + e*x)^6,x]

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 90, normalized size = 0.83 \[ -\frac {2 A e \left (6 a e^2+c \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 B \left (a e^2 (d+5 e x)+c \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^6,x]

[Out]

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

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fricas [A]  time = 0.48, size = 148, normalized size = 1.37 \[ -\frac {30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 2 \, A c d^{2} e + 3 \, B a d e^{2} + 12 \, A a e^{3} + 10 \, {\left (3 \, B c d e^{2} + 2 \, A c e^{3}\right )} x^{2} + 5 \, {\left (3 \, B c d^{2} e + 2 \, A c d e^{2} + 3 \, B a e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/60*(30*B*c*e^3*x^3 + 3*B*c*d^3 + 2*A*c*d^2*e + 3*B*a*d*e^2 + 12*A*a*e^3 + 10*(3*B*c*d*e^2 + 2*A*c*e^3)*x^2
+ 5*(3*B*c*d^2*e + 2*A*c*d*e^2 + 3*B*a*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^
4*e^5*x + d^5*e^4)

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giac [A]  time = 0.18, size = 95, normalized size = 0.88 \[ -\frac {{\left (30 \, B c x^{3} e^{3} + 30 \, B c d x^{2} e^{2} + 15 \, B c d^{2} x e + 3 \, B c d^{3} + 20 \, A c x^{2} e^{3} + 10 \, A c d x e^{2} + 2 \, A c d^{2} e + 15 \, B a x e^{3} + 3 \, B a d e^{2} + 12 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/60*(30*B*c*x^3*e^3 + 30*B*c*d*x^2*e^2 + 15*B*c*d^2*x*e + 3*B*c*d^3 + 20*A*c*x^2*e^3 + 10*A*c*d*x*e^2 + 2*A*
c*d^2*e + 15*B*a*x*e^3 + 3*B*a*d*e^2 + 12*A*a*e^3)*e^(-4)/(x*e + d)^5

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maple [A]  time = 0.06, size = 110, normalized size = 1.02 \[ -\frac {B c}{2 \left (e x +d \right )^{2} e^{4}}-\frac {\left (A e -3 B d \right ) c}{3 \left (e x +d \right )^{3} e^{4}}-\frac {-2 A c d e +B a \,e^{2}+3 B c \,d^{2}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {a A \,e^{3}+A c \,d^{2} e -a B d \,e^{2}-B c \,d^{3}}{5 \left (e x +d \right )^{5} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)/(e*x+d)^6,x)

[Out]

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

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maxima [A]  time = 0.62, size = 148, normalized size = 1.37 \[ -\frac {30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 2 \, A c d^{2} e + 3 \, B a d e^{2} + 12 \, A a e^{3} + 10 \, {\left (3 \, B c d e^{2} + 2 \, A c e^{3}\right )} x^{2} + 5 \, {\left (3 \, B c d^{2} e + 2 \, A c d e^{2} + 3 \, B a e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/60*(30*B*c*e^3*x^3 + 3*B*c*d^3 + 2*A*c*d^2*e + 3*B*a*d*e^2 + 12*A*a*e^3 + 10*(3*B*c*d*e^2 + 2*A*c*e^3)*x^2
+ 5*(3*B*c*d^2*e + 2*A*c*d*e^2 + 3*B*a*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^
4*e^5*x + d^5*e^4)

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mupad [B]  time = 0.07, size = 145, normalized size = 1.34 \[ -\frac {\frac {3\,B\,c\,d^3+2\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+12\,A\,a\,e^3}{60\,e^4}+\frac {x\,\left (3\,B\,c\,d^2+2\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{12\,e^3}+\frac {B\,c\,x^3}{2\,e}+\frac {c\,x^2\,\left (2\,A\,e+3\,B\,d\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)*(A + B*x))/(d + e*x)^6,x)

[Out]

-((12*A*a*e^3 + 3*B*c*d^3 + 3*B*a*d*e^2 + 2*A*c*d^2*e)/(60*e^4) + (x*(3*B*a*e^2 + 3*B*c*d^2 + 2*A*c*d*e))/(12*
e^3) + (B*c*x^3)/(2*e) + (c*x^2*(2*A*e + 3*B*d))/(6*e^2))/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d
^2*e^3*x^3 + 5*d^4*e*x)

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sympy [A]  time = 8.41, size = 165, normalized size = 1.53 \[ \frac {- 12 A a e^{3} - 2 A c d^{2} e - 3 B a d e^{2} - 3 B c d^{3} - 30 B c e^{3} x^{3} + x^{2} \left (- 20 A c e^{3} - 30 B c d e^{2}\right ) + x \left (- 10 A c d e^{2} - 15 B a e^{3} - 15 B c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)/(e*x+d)**6,x)

[Out]

(-12*A*a*e**3 - 2*A*c*d**2*e - 3*B*a*d*e**2 - 3*B*c*d**3 - 30*B*c*e**3*x**3 + x**2*(-20*A*c*e**3 - 30*B*c*d*e*
*2) + x*(-10*A*c*d*e**2 - 15*B*a*e**3 - 15*B*c*d**2*e))/(60*d**5*e**4 + 300*d**4*e**5*x + 600*d**3*e**6*x**2 +
600*d**2*e**7*x**3 + 300*d*e**8*x**4 + 60*e**9*x**5)

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