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

Optimal. Leaf size=197 \[ -\frac {c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac {2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^3}+\frac {c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6} \]

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Rubi [A]  time = 0.15, antiderivative size = 197, 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 {c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 (d+e x)^2}+\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 )}{3 e^6 (d+e x)^3}-\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 )}{4 e^6 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^5}+\frac {c^2 (5 B d-A e)}{e^6 (d+e x)}+\frac {B c^2 \log (d+e x)}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.11, size = 212, normalized size = 1.08 \begin {gather*} \frac {-4 A e \left (3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+B \left (-3 a^2 e^4 (d+5 e x)-6 a c e^2 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+c^2 d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*A*e*(3*a^2*e^4 + a*c*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3
*x^3 + 5*e^4*x^4)) + B*(-3*a^2*e^4*(d + 5*e*x) - 6*a*c*e^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + c^2
*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*B*c^2*(d + e*x)^5*Log[d + e*
x])/(60*e^6*(d + e*x)^5)

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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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 365, normalized size = 1.85 \begin {gather*} \frac {137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \, {\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \, {\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \, {\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x + 60 \, {\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/60*(137*B*c^2*d^5 - 12*A*c^2*d^4*e - 6*B*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 - 3*B*a^2*d*e^4 - 12*A*a^2*e^5 + 60*(
5*B*c^2*d*e^4 - A*c^2*e^5)*x^4 + 60*(15*B*c^2*d^2*e^3 - 2*A*c^2*d*e^4 - B*a*c*e^5)*x^3 + 20*(55*B*c^2*d^3*e^2
- 6*A*c^2*d^2*e^3 - 3*B*a*c*d*e^4 - 2*A*a*c*e^5)*x^2 + 5*(125*B*c^2*d^4*e - 12*A*c^2*d^3*e^2 - 6*B*a*c*d^2*e^3
- 4*A*a*c*d*e^4 - 3*B*a^2*e^5)*x + 60*(B*c^2*e^5*x^5 + 5*B*c^2*d*e^4*x^4 + 10*B*c^2*d^2*e^3*x^3 + 10*B*c^2*d^
3*e^2*x^2 + 5*B*c^2*d^4*e*x + B*c^2*d^5)*log(e*x + d))/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*
x^2 + 5*d^4*e^7*x + d^5*e^6)

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giac [A]  time = 0.20, size = 239, normalized size = 1.21 \begin {gather*} B c^{2} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (60 \, {\left (5 \, B c^{2} d e^{3} - A c^{2} e^{4}\right )} x^{4} + 60 \, {\left (15 \, B c^{2} d^{2} e^{2} - 2 \, A c^{2} d e^{3} - B a c e^{4}\right )} x^{3} + 20 \, {\left (55 \, B c^{2} d^{3} e - 6 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} - 2 \, A a c e^{4}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} - 12 \, A c^{2} d^{3} e - 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} - 3 \, B a^{2} e^{4}\right )} x + {\left (137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

B*c^2*e^(-6)*log(abs(x*e + d)) + 1/60*(60*(5*B*c^2*d*e^3 - A*c^2*e^4)*x^4 + 60*(15*B*c^2*d^2*e^2 - 2*A*c^2*d*e
^3 - B*a*c*e^4)*x^3 + 20*(55*B*c^2*d^3*e - 6*A*c^2*d^2*e^2 - 3*B*a*c*d*e^3 - 2*A*a*c*e^4)*x^2 + 5*(125*B*c^2*d
^4 - 12*A*c^2*d^3*e - 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 - 3*B*a^2*e^4)*x + (137*B*c^2*d^5 - 12*A*c^2*d^4*e - 6*B
*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 - 3*B*a^2*d*e^4 - 12*A*a^2*e^5)*e^(-1))*e^(-5)/(x*e + d)^5

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maple [A]  time = 0.05, size = 362, normalized size = 1.84 \begin {gather*} -\frac {A \,a^{2}}{5 \left (e x +d \right )^{5} e}-\frac {2 A a c \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}-\frac {A \,c^{2} d^{4}}{5 \left (e x +d \right )^{5} e^{5}}+\frac {B \,a^{2} d}{5 \left (e x +d \right )^{5} e^{2}}+\frac {2 B a c \,d^{3}}{5 \left (e x +d \right )^{5} e^{4}}+\frac {B \,c^{2} d^{5}}{5 \left (e x +d \right )^{5} e^{6}}+\frac {A a c d}{\left (e x +d \right )^{4} e^{3}}+\frac {A \,c^{2} d^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {B \,a^{2}}{4 \left (e x +d \right )^{4} e^{2}}-\frac {3 B a c \,d^{2}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {5 B \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {2 A a c}{3 \left (e x +d \right )^{3} e^{3}}-\frac {2 A \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{5}}+\frac {2 B a c d}{\left (e x +d \right )^{3} e^{4}}+\frac {10 B \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{6}}+\frac {2 A \,c^{2} d}{\left (e x +d \right )^{2} e^{5}}-\frac {B a c}{\left (e x +d \right )^{2} e^{4}}-\frac {5 B \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{6}}-\frac {A \,c^{2}}{\left (e x +d \right ) e^{5}}+\frac {5 B \,c^{2} d}{\left (e x +d \right ) e^{6}}+\frac {B \,c^{2} \ln \left (e x +d \right )}{e^{6}} \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)^6,x)

[Out]

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

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maxima [A]  time = 0.64, size = 298, normalized size = 1.51 \begin {gather*} \frac {137 \, B c^{2} d^{5} - 12 \, A c^{2} d^{4} e - 6 \, B a c d^{3} e^{2} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 12 \, A a^{2} e^{5} + 60 \, {\left (5 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 60 \, {\left (15 \, B c^{2} d^{2} e^{3} - 2 \, A c^{2} d e^{4} - B a c e^{5}\right )} x^{3} + 20 \, {\left (55 \, B c^{2} d^{3} e^{2} - 6 \, A c^{2} d^{2} e^{3} - 3 \, B a c d e^{4} - 2 \, A a c e^{5}\right )} x^{2} + 5 \, {\left (125 \, B c^{2} d^{4} e - 12 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} - 4 \, A a c d e^{4} - 3 \, B a^{2} e^{5}\right )} x}{60 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac {B c^{2} \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)^6,x, algorithm="maxima")

[Out]

1/60*(137*B*c^2*d^5 - 12*A*c^2*d^4*e - 6*B*a*c*d^3*e^2 - 4*A*a*c*d^2*e^3 - 3*B*a^2*d*e^4 - 12*A*a^2*e^5 + 60*(
5*B*c^2*d*e^4 - A*c^2*e^5)*x^4 + 60*(15*B*c^2*d^2*e^3 - 2*A*c^2*d*e^4 - B*a*c*e^5)*x^3 + 20*(55*B*c^2*d^3*e^2
- 6*A*c^2*d^2*e^3 - 3*B*a*c*d*e^4 - 2*A*a*c*e^5)*x^2 + 5*(125*B*c^2*d^4*e - 12*A*c^2*d^3*e^2 - 6*B*a*c*d^2*e^3
- 4*A*a*c*d*e^4 - 3*B*a^2*e^5)*x)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x +
d^5*e^6) + B*c^2*log(e*x + d)/e^6

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mupad [B]  time = 1.78, size = 243, normalized size = 1.23 \begin {gather*} \frac {B\,c^2\,\ln \left (d+e\,x\right )}{e^6}-\frac {x^2\,\left (-\frac {55\,B\,c^2\,d^3\,e^2}{3}+2\,A\,c^2\,d^2\,e^3+B\,a\,c\,d\,e^4+\frac {2\,A\,a\,c\,e^5}{3}\right )+x^3\,\left (-15\,B\,c^2\,d^2\,e^3+2\,A\,c^2\,d\,e^4+B\,a\,c\,e^5\right )+x^4\,\left (A\,c^2\,e^5-5\,B\,c^2\,d\,e^4\right )+x\,\left (\frac {B\,a^2\,e^5}{4}+\frac {B\,a\,c\,d^2\,e^3}{2}+\frac {A\,a\,c\,d\,e^4}{3}-\frac {125\,B\,c^2\,d^4\,e}{12}+A\,c^2\,d^3\,e^2\right )+\frac {A\,a^2\,e^5}{5}-\frac {137\,B\,c^2\,d^5}{60}+\frac {B\,a^2\,d\,e^4}{20}+\frac {A\,c^2\,d^4\,e}{5}+\frac {A\,a\,c\,d^2\,e^3}{15}+\frac {B\,a\,c\,d^3\,e^2}{10}}{e^6\,{\left (d+e\,x\right )}^5} \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)^6,x)

[Out]

(B*c^2*log(d + e*x))/e^6 - (x^2*((2*A*a*c*e^5)/3 + 2*A*c^2*d^2*e^3 - (55*B*c^2*d^3*e^2)/3 + B*a*c*d*e^4) + x^3
*(B*a*c*e^5 + 2*A*c^2*d*e^4 - 15*B*c^2*d^2*e^3) + x^4*(A*c^2*e^5 - 5*B*c^2*d*e^4) + x*((B*a^2*e^5)/4 - (125*B*
c^2*d^4*e)/12 + A*c^2*d^3*e^2 + (A*a*c*d*e^4)/3 + (B*a*c*d^2*e^3)/2) + (A*a^2*e^5)/5 - (137*B*c^2*d^5)/60 + (B
*a^2*d*e^4)/20 + (A*c^2*d^4*e)/5 + (A*a*c*d^2*e^3)/15 + (B*a*c*d^3*e^2)/10)/(e^6*(d + e*x)^5)

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sympy [A]  time = 54.06, size = 326, normalized size = 1.65 \begin {gather*} \frac {B c^{2} \log {\left (d + e x \right )}}{e^{6}} + \frac {- 12 A a^{2} e^{5} - 4 A a c d^{2} e^{3} - 12 A c^{2} d^{4} e - 3 B a^{2} d e^{4} - 6 B a c d^{3} e^{2} + 137 B c^{2} d^{5} + x^{4} \left (- 60 A c^{2} e^{5} + 300 B c^{2} d e^{4}\right ) + x^{3} \left (- 120 A c^{2} d e^{4} - 60 B a c e^{5} + 900 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 40 A a c e^{5} - 120 A c^{2} d^{2} e^{3} - 60 B a c d e^{4} + 1100 B c^{2} d^{3} e^{2}\right ) + x \left (- 20 A a c d e^{4} - 60 A c^{2} d^{3} e^{2} - 15 B a^{2} e^{5} - 30 B a c d^{2} e^{3} + 625 B c^{2} d^{4} e\right )}{60 d^{5} e^{6} + 300 d^{4} e^{7} x + 600 d^{3} e^{8} x^{2} + 600 d^{2} e^{9} x^{3} + 300 d e^{10} x^{4} + 60 e^{11} x^{5}} \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)**6,x)

[Out]

B*c**2*log(d + e*x)/e**6 + (-12*A*a**2*e**5 - 4*A*a*c*d**2*e**3 - 12*A*c**2*d**4*e - 3*B*a**2*d*e**4 - 6*B*a*c
*d**3*e**2 + 137*B*c**2*d**5 + x**4*(-60*A*c**2*e**5 + 300*B*c**2*d*e**4) + x**3*(-120*A*c**2*d*e**4 - 60*B*a*
c*e**5 + 900*B*c**2*d**2*e**3) + x**2*(-40*A*a*c*e**5 - 120*A*c**2*d**2*e**3 - 60*B*a*c*d*e**4 + 1100*B*c**2*d
**3*e**2) + x*(-20*A*a*c*d*e**4 - 60*A*c**2*d**3*e**2 - 15*B*a**2*e**5 - 30*B*a*c*d**2*e**3 + 625*B*c**2*d**4*
e))/(60*d**5*e**6 + 300*d**4*e**7*x + 600*d**3*e**8*x**2 + 600*d**2*e**9*x**3 + 300*d*e**10*x**4 + 60*e**11*x*
*5)

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