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

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

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Rubi [A]  time = 0.19, antiderivative size = 189, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.11, size = 232, normalized size = 1.23 \begin {gather*} \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 (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\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 (47 d^5+81 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\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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.41, size = 411, normalized size = 2.17 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.15, size = 237, normalized size = 1.25 \begin {gather*} 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 {gather*}

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

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maple [A]  time = 0.06, size = 346, normalized size = 1.83 \begin {gather*} -\frac {A \,a^{2}}{3 \left (e x +d \right )^{3} e}-\frac {2 A a c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A \,c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}+\frac {B \,a^{2} d}{3 \left (e x +d \right )^{3} e^{2}}+\frac {2 B a c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}+\frac {B \,c^{2} d^{5}}{3 \left (e x +d \right )^{3} e^{6}}+\frac {2 A a c d}{\left (e x +d \right )^{2} e^{3}}+\frac {2 A \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {B \,a^{2}}{2 \left (e x +d \right )^{2} e^{2}}-\frac {3 B a c \,d^{2}}{\left (e x +d \right )^{2} e^{4}}-\frac {5 B \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{6}}+\frac {B \,c^{2} x^{2}}{2 e^{4}}-\frac {2 A a c}{\left (e x +d \right ) e^{3}}-\frac {6 A \,c^{2} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {A \,c^{2} x}{e^{4}}+\frac {6 B a c d}{\left (e x +d \right ) e^{4}}+\frac {2 B a c \ln \left (e x +d \right )}{e^{4}}+\frac {10 B \,c^{2} d^{3}}{\left (e x +d \right ) e^{6}}+\frac {10 B \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,c^{2} d x}{e^{5}} \end {gather*}

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

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maxima [A]  time = 0.61, size = 270, normalized size = 1.43 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.77, size = 268, normalized size = 1.42 \begin {gather*} x\,\left (\frac {A\,c^2}{e^4}-\frac {4\,B\,c^2\,d}{e^5}\right )-\frac {x\,\left (\frac {B\,a^2\,e^4}{2}-9\,B\,a\,c\,d^2\,e^2+2\,A\,a\,c\,d\,e^3-\frac {35\,B\,c^2\,d^4}{2}+10\,A\,c^2\,d^3\,e\right )+\frac {B\,a^2\,d\,e^4+2\,A\,a^2\,e^5-22\,B\,a\,c\,d^3\,e^2+4\,A\,a\,c\,d^2\,e^3-47\,B\,c^2\,d^5+26\,A\,c^2\,d^4\,e}{6\,e}+x^2\,\left (-10\,B\,c^2\,d^3\,e+6\,A\,c^2\,d^2\,e^2-6\,B\,a\,c\,d\,e^3+2\,A\,a\,c\,e^4\right )}{d^3\,e^5+3\,d^2\,e^6\,x+3\,d\,e^7\,x^2+e^8\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (10\,B\,c^2\,d^2-4\,A\,c^2\,d\,e+2\,B\,a\,c\,e^2\right )}{e^6}+\frac {B\,c^2\,x^2}{2\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((A*c^2)/e^4 - (4*B*c^2*d)/e^5) - (x*((B*a^2*e^4)/2 - (35*B*c^2*d^4)/2 + 10*A*c^2*d^3*e + 2*A*a*c*d*e^3 - 9*
B*a*c*d^2*e^2) + (2*A*a^2*e^5 - 47*B*c^2*d^5 + B*a^2*d*e^4 + 26*A*c^2*d^4*e + 4*A*a*c*d^2*e^3 - 22*B*a*c*d^3*e
^2)/(6*e) + x^2*(2*A*a*c*e^4 - 10*B*c^2*d^3*e + 6*A*c^2*d^2*e^2 - 6*B*a*c*d*e^3))/(d^3*e^5 + e^8*x^3 + 3*d^2*e
^6*x + 3*d*e^7*x^2) + (log(d + e*x)*(10*B*c^2*d^2 + 2*B*a*c*e^2 - 4*A*c^2*d*e))/e^6 + (B*c^2*x^2)/(2*e^4)

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sympy [A]  time = 9.16, size = 294, normalized size = 1.56 \begin {gather*} \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}} + x \left (\frac {A c^{2}}{e^{4}} - \frac {4 B c^{2} d}{e^{5}}\right ) + \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}} \end {gather*}

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

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