Optimal. Leaf size=198 \[ \frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.357378, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1247, 711, 68} \[ \frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1247
Rule 711
Rule 68
Rubi steps
\begin{align*} \int \frac{x \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(d+e x)^q}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{2 c (d+e x)^q}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}-\frac{2 c (d+e x)^q}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{c \operatorname{Subst}\left (\int \frac{(d+e x)^q}{b-\sqrt{b^2-4 a c}+2 c x} \, dx,x,x^2\right )}{\sqrt{b^2-4 a c}}-\frac{c \operatorname{Subst}\left (\int \frac{(d+e x)^q}{b+\sqrt{b^2-4 a c}+2 c x} \, dx,x,x^2\right )}{\sqrt{b^2-4 a c}}\\ &=-\frac{c \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac{2 c \left (d+e x^2\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) (1+q)}+\frac{c \left (d+e x^2\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac{2 c \left (d+e x^2\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1+q)}\\ \end{align*}
Mathematica [A] time = 0.299574, size = 168, normalized size = 0.85 \[ \frac{c \left (d+e x^2\right )^{q+1} \left (\frac{\, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}-\frac{\, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}\right )}{(q+1) \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \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}{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}{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}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]