Optimal. Leaf size=273 \[ -\frac{x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (b-\sqrt{b^2-4 a c}\right )}-\frac{x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.53064, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1303, 246, 245, 1692, 430, 429} \[ -\frac{x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (b-\sqrt{b^2-4 a c}\right )}-\frac{x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1303
Rule 246
Rule 245
Rule 1692
Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{x^4 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\int \left (\frac{\left (d+e x^2\right )^q}{c}-\frac{\left (a+b x^2\right ) \left (d+e x^2\right )^q}{c \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac{\int \left (d+e x^2\right )^q \, dx}{c}-\frac{\int \frac{\left (a+b x^2\right ) \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx}{c}\\ &=-\frac{\int \left (\frac{\left (b+\frac{-b^2+2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2}+\frac{\left (b-\frac{-b^2+2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2}\right ) \, dx}{c}+\frac{\left (\left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \left (1+\frac{e x^2}{d}\right )^q \, dx}{c}\\ &=\frac{x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c}-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \int \frac{\left (d+e x^2\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{c}-\frac{\left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \int \frac{\left (d+e x^2\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{c}\\ &=\frac{x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c}-\frac{\left (\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{c}-\frac{\left (\left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q}\right ) \int \frac{\left (1+\frac{e x^2}{d}\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{c}\\ &=-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (b-\sqrt{b^2-4 a c}\right )}-\frac{\left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (b+\sqrt{b^2-4 a c}\right )}+\frac{x \left (d+e x^2\right )^q \left (1+\frac{e x^2}{d}\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c}\\ \end{align*}
Mathematica [F] time = 0.251013, size = 0, normalized size = 0. \[ \int \frac{x^4 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4} \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{q} x^{4}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q} x^{4}}{c x^{4} + b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{q} x^{4}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]