3.32.10 \(\int \frac {(1+x^2) (-a-b x+a x^2)}{x^2 (-c+d x+c x^2) \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}}} \, dx\)

Optimal. Leaf size=622 \[ \frac {\log \left (-1+\sqrt [3]{\frac {x^2-c_0 x-1}{x^2-c_1 x-1}}\right ) (-a c c_0+a c c_1+3 a d+3 b c)}{3 c^2}+\frac {\log \left (\left (\frac {x^2-c_0 x-1}{x^2-c_1 x-1}\right ){}^{2/3}+\sqrt [3]{\frac {x^2-c_0 x-1}{x^2-c_1 x-1}}+1\right ) (a c c_0-a c c_1-3 a d-3 b c)}{6 c^2}+\frac {\sqrt [3]{-d-c c_1} (a d+b c) \log \left (\sqrt [3]{\frac {-x^2+c_0 x+1}{-x^2+c_1 x+1}} \sqrt [3]{-d-c c_1}+\sqrt [3]{d+c c_0}\right )}{c^2 \sqrt [3]{d+c c_0}}-\frac {\sqrt [3]{-d-c c_1} (a d+b c) \log \left (-\sqrt [3]{\frac {-x^2+c_0 x+1}{-x^2+c_1 x+1}} \sqrt [3]{-d-c c_1} \sqrt [3]{d+c c_0}+\left (\frac {-x^2+c_0 x+1}{-x^2+c_1 x+1}\right ){}^{2/3} (-d-c c_1){}^{2/3}+(d+c c_0){}^{2/3}\right )}{2 c^2 \sqrt [3]{d+c c_0}}+\frac {\sqrt {3} \sqrt [3]{-d-c c_1} (a d+b c) \tan ^{-1}\left (\frac {\sqrt [3]{d+c c_0}-2 \sqrt [3]{\frac {-x^2+c_0 x+1}{-x^2+c_1 x+1}} \sqrt [3]{-d-c c_1}}{\sqrt {3} \sqrt [3]{d+c c_0}}\right )}{c^2 \sqrt [3]{d+c c_0}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{\frac {x^2-c_0 x-1}{x^2-c_1 x-1}}}{\sqrt {3}}\right ) \left (-\sqrt {3} a c c_0+\sqrt {3} a c c_1+3 \sqrt {3} a d+3 \sqrt {3} b c\right )}{3 c^2}+\frac {\left (\frac {x^2-c_0 x-1}{x^2-c_1 x-1}\right ){}^{2/3} \left (a x^2-a c_1 x-a\right )}{c x} \]

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Rubi [F]  time = 13.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^2\right ) \left (-a-b x+a x^2\right )}{x^2 \left (-c+d x+c x^2\right ) \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^2)*(-a - b*x + a*x^2))/(x^2*(-c + d*x + c*x^2)*((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3)),x]

[Out]

(a*(-1 + x^2 - x*C[0])^(1/3)*Defer[Int][(-1 + x^2 - x*C[1])^(1/3)/(-1 + x^2 - x*C[0])^(1/3), x])/(c*(-1 + x^2
- x*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3)) + (a*(-1 + x^2 - x*C[0])^(1/3)*Defer[Int][(-1 +
x^2 - x*C[1])^(1/3)/(x^2*(-1 + x^2 - x*C[0])^(1/3)), x])/(c*(-1 + x^2 - x*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1
- x^2 + x*C[1]))^(1/3)) + ((b*c + a*d)*(-1 + x^2 - x*C[0])^(1/3)*Defer[Int][(-1 + x^2 - x*C[1])^(1/3)/(x*(-1 +
x^2 - x*C[0])^(1/3)), x])/(c^2*(-1 + x^2 - x*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3)) - (2*
(b*c + a*d)*(-1 + x^2 - x*C[0])^(1/3)*Defer[Int][(-1 + x^2 - x*C[1])^(1/3)/((d - Sqrt[4*c^2 + d^2] + 2*c*x)*(-
1 + x^2 - x*C[0])^(1/3)), x])/(c*(-1 + x^2 - x*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3)) - (2
*(b*c + a*d)*(-1 + x^2 - x*C[0])^(1/3)*Defer[Int][(-1 + x^2 - x*C[1])^(1/3)/((d + Sqrt[4*c^2 + d^2] + 2*c*x)*(
-1 + x^2 - x*C[0])^(1/3)), x])/(c*(-1 + x^2 - x*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right ) \left (-a-b x+a x^2\right )}{x^2 \left (-c+d x+c x^2\right ) \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}}} \, dx &=\frac {\sqrt [3]{-1+x^2-x c_0} \int \frac {\left (1+x^2\right ) \left (-a-b x+a x^2\right ) \sqrt [3]{-1+x^2-x c_1}}{x^2 \left (-c+d x+c x^2\right ) \sqrt [3]{-1+x^2-x c_0}} \, dx}{\sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}\\ &=\frac {\sqrt [3]{-1+x^2-x c_0} \int \left (\frac {a \sqrt [3]{-1+x^2-x c_1}}{c \sqrt [3]{-1+x^2-x c_0}}+\frac {a \sqrt [3]{-1+x^2-x c_1}}{c x^2 \sqrt [3]{-1+x^2-x c_0}}+\frac {(b c+a d) \sqrt [3]{-1+x^2-x c_1}}{c^2 x \sqrt [3]{-1+x^2-x c_0}}-\frac {(b c+a d) (d+2 c x) \sqrt [3]{-1+x^2-x c_1}}{c^2 \left (-c+d x+c x^2\right ) \sqrt [3]{-1+x^2-x c_0}}\right ) \, dx}{\sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}\\ &=\frac {\left (a \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{\sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}+\frac {\left (a \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{x^2 \sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}+\frac {\left ((b c+a d) \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{x \sqrt [3]{-1+x^2-x c_0}} \, dx}{c^2 \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}-\frac {\left ((b c+a d) \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {(d+2 c x) \sqrt [3]{-1+x^2-x c_1}}{\left (-c+d x+c x^2\right ) \sqrt [3]{-1+x^2-x c_0}} \, dx}{c^2 \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}\\ &=\frac {\left (a \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{\sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}+\frac {\left (a \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{x^2 \sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}+\frac {\left ((b c+a d) \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{x \sqrt [3]{-1+x^2-x c_0}} \, dx}{c^2 \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}-\frac {\left ((b c+a d) \sqrt [3]{-1+x^2-x c_0}\right ) \int \left (\frac {2 c \sqrt [3]{-1+x^2-x c_1}}{\left (d-\sqrt {4 c^2+d^2}+2 c x\right ) \sqrt [3]{-1+x^2-x c_0}}+\frac {2 c \sqrt [3]{-1+x^2-x c_1}}{\left (d+\sqrt {4 c^2+d^2}+2 c x\right ) \sqrt [3]{-1+x^2-x c_0}}\right ) \, dx}{c^2 \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}\\ &=\frac {\left (a \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{\sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}+\frac {\left (a \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{x^2 \sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}+\frac {\left ((b c+a d) \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{x \sqrt [3]{-1+x^2-x c_0}} \, dx}{c^2 \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}-\frac {\left (2 (b c+a d) \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{\left (d-\sqrt {4 c^2+d^2}+2 c x\right ) \sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}-\frac {\left (2 (b c+a d) \sqrt [3]{-1+x^2-x c_0}\right ) \int \frac {\sqrt [3]{-1+x^2-x c_1}}{\left (d+\sqrt {4 c^2+d^2}+2 c x\right ) \sqrt [3]{-1+x^2-x c_0}} \, dx}{c \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}} \sqrt [3]{-1+x^2-x c_1}}\\ \end {align*}

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Mathematica [F]  time = 3.90, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2\right ) \left (-a-b x+a x^2\right )}{x^2 \left (-c+d x+c x^2\right ) \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^2)*(-a - b*x + a*x^2))/(x^2*(-c + d*x + c*x^2)*((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/
3)),x]

[Out]

Integrate[((1 + x^2)*(-a - b*x + a*x^2))/(x^2*(-c + d*x + c*x^2)*((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/
3)), x]

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IntegrateAlgebraic [A]  time = 2.91, size = 622, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} (b c+a d) \tan ^{-1}\left (\frac {\sqrt [3]{d+c c_0}-2 \sqrt [3]{-d-c c_1} \sqrt [3]{\frac {1-x^2+x c_0}{1-x^2+x c_1}}}{\sqrt {3} \sqrt [3]{d+c c_0}}\right ) \sqrt [3]{-d-c c_1}}{c^2 \sqrt [3]{d+c c_0}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}}}{\sqrt {3}}\right ) \left (3 \sqrt {3} b c+3 \sqrt {3} a d-\sqrt {3} a c c_0+\sqrt {3} a c c_1\right )}{3 c^2}+\frac {\left (\frac {-1+x^2-x c_0}{-1+x^2-x c_1}\right ){}^{2/3} \left (-a+a x^2-a x c_1\right )}{c x}+\frac {(3 b c+3 a d-a c c_0+a c c_1) \log \left (-1+\sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}}\right )}{3 c^2}+\frac {(-3 b c-3 a d+a c c_0-a c c_1) \log \left (1+\sqrt [3]{\frac {-1+x^2-x c_0}{-1+x^2-x c_1}}+\left (\frac {-1+x^2-x c_0}{-1+x^2-x c_1}\right ){}^{2/3}\right )}{6 c^2}+\frac {(b c+a d) \sqrt [3]{-d-c c_1} \log \left (\sqrt [3]{d+c c_0}+\sqrt [3]{-d-c c_1} \sqrt [3]{\frac {1-x^2+x c_0}{1-x^2+x c_1}}\right )}{c^2 \sqrt [3]{d+c c_0}}-\frac {(b c+a d) \sqrt [3]{-d-c c_1} \log \left ((d+c c_0){}^{2/3}-\sqrt [3]{d+c c_0} \sqrt [3]{-d-c c_1} \sqrt [3]{\frac {1-x^2+x c_0}{1-x^2+x c_1}}+(-d-c c_1){}^{2/3} \left (\frac {1-x^2+x c_0}{1-x^2+x c_1}\right ){}^{2/3}\right )}{2 c^2 \sqrt [3]{d+c c_0}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^2)*(-a - b*x + a*x^2))/(x^2*(-c + d*x + c*x^2)*((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C
[1]))^(1/3)),x]

[Out]

(Sqrt[3]*(b*c + a*d)*ArcTan[((d + c*C[0])^(1/3) - 2*(-d - c*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1])
)^(1/3))/(Sqrt[3]*(d + c*C[0])^(1/3))]*(-d - c*C[1])^(1/3))/(c^2*(d + c*C[0])^(1/3)) + (ArcTan[(1 + 2*((-1 + x
^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3))/Sqrt[3]]*(3*Sqrt[3]*b*c + 3*Sqrt[3]*a*d - Sqrt[3]*a*c*C[0] + Sqrt[3]*
a*c*C[1]))/(3*c^2) + (((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(2/3)*(-a + a*x^2 - a*x*C[1]))/(c*x) + ((3*b*c
+ 3*a*d - a*c*C[0] + a*c*C[1])*Log[-1 + ((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3)])/(3*c^2) + ((-3*b*c
- 3*a*d + a*c*C[0] - a*c*C[1])*Log[1 + ((-1 + x^2 - x*C[0])/(-1 + x^2 - x*C[1]))^(1/3) + ((-1 + x^2 - x*C[0])/
(-1 + x^2 - x*C[1]))^(2/3)])/(6*c^2) + ((b*c + a*d)*(-d - c*C[1])^(1/3)*Log[(d + c*C[0])^(1/3) + (-d - c*C[1])
^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(1/3)])/(c^2*(d + c*C[0])^(1/3)) - ((b*c + a*d)*(-d - c*C[1])^(
1/3)*Log[(d + c*C[0])^(2/3) - (d + c*C[0])^(1/3)*(-d - c*C[1])^(1/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(
1/3) + (-d - c*C[1])^(2/3)*((1 - x^2 + x*C[0])/(1 - x^2 + x*C[1]))^(2/3)])/(2*c^2*(d + c*C[0])^(1/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(a*x^2-b*x-a)/x^2/(c*x^2+d*x-c)/((-_C0*x+x^2-1)/(-_C1*x+x^2-1))^(1/3),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} - b x - a\right )} {\left (x^{2} + 1\right )}}{{\left (c x^{2} + d x - c\right )} x^{2} \left (\frac {\_{C_{0}} x - x^{2} + 1}{\_{C_{1}} x - x^{2} + 1}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(a*x^2-b*x-a)/x^2/(c*x^2+d*x-c)/((-_C0*x+x^2-1)/(-_C1*x+x^2-1))^(1/3),x, algorithm="giac")

[Out]

integrate((a*x^2 - b*x - a)*(x^2 + 1)/((c*x^2 + d*x - c)*x^2*((_C0*x - x^2 + 1)/(_C1*x - x^2 + 1))^(1/3)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+1\right ) \left (a \,x^{2}-b x -a \right )}{x^{2} \left (c \,x^{2}+d x -c \right ) \left (\frac {-\textit {\_C0} x +x^{2}-1}{-\textit {\_C1} x +x^{2}-1}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)*(a*x^2-b*x-a)/x^2/(c*x^2+d*x-c)/((-_C0*x+x^2-1)/(-_C1*x+x^2-1))^(1/3),x)

[Out]

int((x^2+1)*(a*x^2-b*x-a)/x^2/(c*x^2+d*x-c)/((-_C0*x+x^2-1)/(-_C1*x+x^2-1))^(1/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{2} - b x - a\right )} {\left (x^{2} + 1\right )}}{{\left (c x^{2} + d x - c\right )} x^{2} \left (\frac {\_{C_{0}} x - x^{2} + 1}{\_{C_{1}} x - x^{2} + 1}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)*(a*x^2-b*x-a)/x^2/(c*x^2+d*x-c)/((-_C0*x+x^2-1)/(-_C1*x+x^2-1))^(1/3),x, algorithm="maxima")

[Out]

integrate((a*x^2 - b*x - a)*(x^2 + 1)/((c*x^2 + d*x - c)*x^2*((_C0*x - x^2 + 1)/(_C1*x - x^2 + 1))^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (x^2+1\right )\,\left (-a\,x^2+b\,x+a\right )}{x^2\,{\left (\frac {-x^2+_{\mathrm {C0}}\,x+1}{-x^2+_{\mathrm {C1}}\,x+1}\right )}^{1/3}\,\left (c\,x^2+d\,x-c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^2 + 1)*(a + b*x - a*x^2))/(x^2*((_C0*x - x^2 + 1)/(_C1*x - x^2 + 1))^(1/3)*(d*x - c + c*x^2)),x)

[Out]

int(-((x^2 + 1)*(a + b*x - a*x^2))/(x^2*((_C0*x - x^2 + 1)/(_C1*x - x^2 + 1))^(1/3)*(d*x - c + c*x^2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)*(a*x**2-b*x-a)/x**2/(c*x**2+d*x-c)/((-_C0*x+x**2-1)/(-_C1*x+x**2-1))**(1/3),x)

[Out]

Timed out

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