3.1306 \(\int \frac{(A+B x) (a+c x^2)^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=189 \[ \frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{2 e^6 (d+e x)^2}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6 (d+e x)^3}+\frac{2 c \log (d+e x) \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}-\frac{c^2 x (4 B d-A e)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]

[Out]

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

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Rubi [A]  time = 0.190121, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 (d+e x)}-\frac{\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{2 e^6 (d+e x)^2}+\frac{\left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6 (d+e x)^3}+\frac{2 c \log (d+e x) \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}-\frac{c^2 x (4 B d-A e)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.117, size = 232, normalized size = 1.23 \[ \frac{-2 A e \left (a^2 e^4+2 a c e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+c^2 \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )\right )+B \left (-a^2 e^4 (d+3 e x)+2 a c d e^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )+c^2 \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )\right )+12 c (d+e x)^3 \log (d+e x) \left (a B e^2-2 A c d e+5 B c d^2\right )}{6 e^6 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^4,x]

[Out]

(-2*A*e*(a^2*e^4 + 2*a*c*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2) + c^2*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*
x^3 - 3*e^4*x^4)) + B*(-(a^2*e^4*(d + 3*e*x)) + 2*a*c*d*e^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + c^2*(47*d^5 + 8
1*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5)) + 12*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^
2)*(d + e*x)^3*Log[d + e*x])/(6*e^6*(d + e*x)^3)

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Maple [A]  time = 0.01, size = 346, normalized size = 1.8 \begin{align*}{\frac{B{c}^{2}{x}^{2}}{2\,{e}^{4}}}+{\frac{A{c}^{2}x}{{e}^{4}}}-4\,{\frac{B{c}^{2}dx}{{e}^{5}}}-2\,{\frac{aAc}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+6\,{\frac{aBcd}{{e}^{4} \left ( ex+d \right ) }}+10\,{\frac{B{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}+2\,{\frac{Adac}{{e}^{3} \left ( ex+d \right ) ^{2}}}+2\,{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{B{a}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-3\,{\frac{aBc{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{5\,B{c}^{2}{d}^{4}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{A{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{2\,A{d}^{2}ac}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{A{d}^{4}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{Bd{a}^{2}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{2\,B{d}^{3}ac}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{B{c}^{2}{d}^{5}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-4\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}d}{{e}^{5}}}+2\,{\frac{c\ln \left ( ex+d \right ) aB}{{e}^{4}}}+10\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{2}}{{e}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^4,x)

[Out]

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

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Maxima [A]  time = 1.02078, size = 365, normalized size = 1.93 \begin{align*} \frac{47 \, B c^{2} d^{5} - 26 \, A c^{2} d^{4} e + 22 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 2 \, A a^{2} e^{5} + 12 \,{\left (5 \, B c^{2} d^{3} e^{2} - 3 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} - A a c e^{5}\right )} x^{2} + 3 \,{\left (35 \, B c^{2} d^{4} e - 20 \, A c^{2} d^{3} e^{2} + 18 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - B a^{2} e^{5}\right )} x}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{B c^{2} e x^{2} - 2 \,{\left (4 \, B c^{2} d - A c^{2} e\right )} x}{2 \, e^{5}} + \frac{2 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(47*B*c^2*d^5 - 26*A*c^2*d^4*e + 22*B*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 - B*a^2*d*e^4 - 2*A*a^2*e^5 + 12*(5*B*
c^2*d^3*e^2 - 3*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 - A*a*c*e^5)*x^2 + 3*(35*B*c^2*d^4*e - 20*A*c^2*d^3*e^2 + 18*B*a
*c*d^2*e^3 - 4*A*a*c*d*e^4 - B*a^2*e^5)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6) + 1/2*(B*c^2*e*x^2
- 2*(4*B*c^2*d - A*c^2*e)*x)/e^5 + 2*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c*e^2)*log(e*x + d)/e^6

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Fricas [B]  time = 1.76558, size = 867, normalized size = 4.59 \begin{align*} \frac{3 \, B c^{2} e^{5} x^{5} + 47 \, B c^{2} d^{5} - 26 \, A c^{2} d^{4} e + 22 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 2 \, A a^{2} e^{5} - 3 \,{\left (5 \, B c^{2} d e^{4} - 2 \, A c^{2} e^{5}\right )} x^{4} - 9 \,{\left (7 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, B c^{2} d^{3} e^{2} + 6 \, A c^{2} d^{2} e^{3} - 12 \, B a c d e^{4} + 4 \, A a c e^{5}\right )} x^{2} + 3 \,{\left (27 \, B c^{2} d^{4} e - 18 \, A c^{2} d^{3} e^{2} + 18 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - B a^{2} e^{5}\right )} x + 12 \,{\left (5 \, B c^{2} d^{5} - 2 \, A c^{2} d^{4} e + B a c d^{3} e^{2} +{\left (5 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 3 \,{\left (5 \, B c^{2} d^{3} e^{2} - 2 \, A c^{2} d^{2} e^{3} + B a c d e^{4}\right )} x^{2} + 3 \,{\left (5 \, B c^{2} d^{4} e - 2 \, A c^{2} d^{3} e^{2} + B a c d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(3*B*c^2*e^5*x^5 + 47*B*c^2*d^5 - 26*A*c^2*d^4*e + 22*B*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 - B*a^2*d*e^4 - 2*A*
a^2*e^5 - 3*(5*B*c^2*d*e^4 - 2*A*c^2*e^5)*x^4 - 9*(7*B*c^2*d^2*e^3 - 2*A*c^2*d*e^4)*x^3 - 3*(3*B*c^2*d^3*e^2 +
6*A*c^2*d^2*e^3 - 12*B*a*c*d*e^4 + 4*A*a*c*e^5)*x^2 + 3*(27*B*c^2*d^4*e - 18*A*c^2*d^3*e^2 + 18*B*a*c*d^2*e^3
- 4*A*a*c*d*e^4 - B*a^2*e^5)*x + 12*(5*B*c^2*d^5 - 2*A*c^2*d^4*e + B*a*c*d^3*e^2 + (5*B*c^2*d^2*e^3 - 2*A*c^2
*d*e^4 + B*a*c*e^5)*x^3 + 3*(5*B*c^2*d^3*e^2 - 2*A*c^2*d^2*e^3 + B*a*c*d*e^4)*x^2 + 3*(5*B*c^2*d^4*e - 2*A*c^2
*d^3*e^2 + B*a*c*d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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Sympy [A]  time = 35.0825, size = 292, normalized size = 1.54 \begin{align*} \frac{B c^{2} x^{2}}{2 e^{4}} + \frac{2 c \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} + \frac{- 2 A a^{2} e^{5} - 4 A a c d^{2} e^{3} - 26 A c^{2} d^{4} e - B a^{2} d e^{4} + 22 B a c d^{3} e^{2} + 47 B c^{2} d^{5} + x^{2} \left (- 12 A a c e^{5} - 36 A c^{2} d^{2} e^{3} + 36 B a c d e^{4} + 60 B c^{2} d^{3} e^{2}\right ) + x \left (- 12 A a c d e^{4} - 60 A c^{2} d^{3} e^{2} - 3 B a^{2} e^{5} + 54 B a c d^{2} e^{3} + 105 B c^{2} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} - \frac{x \left (- A c^{2} e + 4 B c^{2} d\right )}{e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**4,x)

[Out]

B*c**2*x**2/(2*e**4) + 2*c*(-2*A*c*d*e + B*a*e**2 + 5*B*c*d**2)*log(d + e*x)/e**6 + (-2*A*a**2*e**5 - 4*A*a*c*
d**2*e**3 - 26*A*c**2*d**4*e - B*a**2*d*e**4 + 22*B*a*c*d**3*e**2 + 47*B*c**2*d**5 + x**2*(-12*A*a*c*e**5 - 36
*A*c**2*d**2*e**3 + 36*B*a*c*d*e**4 + 60*B*c**2*d**3*e**2) + x*(-12*A*a*c*d*e**4 - 60*A*c**2*d**3*e**2 - 3*B*a
**2*e**5 + 54*B*a*c*d**2*e**3 + 105*B*c**2*d**4*e))/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x*
*3) - x*(-A*c**2*e + 4*B*c**2*d)/e**5

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Giac [A]  time = 1.30727, size = 320, normalized size = 1.69 \begin{align*} 2 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B c^{2} x^{2} e^{4} - 8 \, B c^{2} d x e^{3} + 2 \, A c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, B c^{2} d^{5} - 26 \, A c^{2} d^{4} e + 22 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - B a^{2} d e^{4} - 2 \, A a^{2} e^{5} + 12 \,{\left (5 \, B c^{2} d^{3} e^{2} - 3 \, A c^{2} d^{2} e^{3} + 3 \, B a c d e^{4} - A a c e^{5}\right )} x^{2} + 3 \,{\left (35 \, B c^{2} d^{4} e - 20 \, A c^{2} d^{3} e^{2} + 18 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - B a^{2} e^{5}\right )} x\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^4,x, algorithm="giac")

[Out]

2*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c*e^2)*e^(-6)*log(abs(x*e + d)) + 1/2*(B*c^2*x^2*e^4 - 8*B*c^2*d*x*e^3 + 2*
A*c^2*x*e^4)*e^(-8) + 1/6*(47*B*c^2*d^5 - 26*A*c^2*d^4*e + 22*B*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 - B*a^2*d*e^4 -
2*A*a^2*e^5 + 12*(5*B*c^2*d^3*e^2 - 3*A*c^2*d^2*e^3 + 3*B*a*c*d*e^4 - A*a*c*e^5)*x^2 + 3*(35*B*c^2*d^4*e - 20*
A*c^2*d^3*e^2 + 18*B*a*c*d^2*e^3 - 4*A*a*c*d*e^4 - B*a^2*e^5)*x)*e^(-6)/(x*e + d)^3