Optimal. Leaf size=210 \[ -\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{2 (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
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Rubi [A] time = 0.328882, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1251, 830, 68} \[ -\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{2 (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 830
Rule 68
Rubi steps
\begin{align*} \int \frac{x^3 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (d+e x)^q}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) (d+e x)^q}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) (d+e x)^q}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^q}{b-\sqrt{b^2-4 a c}+2 c x} \, dx,x,x^2\right )+\frac{1}{2} \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^q}{b+\sqrt{b^2-4 a c}+2 c x} \, dx,x,x^2\right )\\ &=-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \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 )}{2 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) (1+q)}-\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \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 )}{2 \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) (1+q)}\\ \end{align*}
Mathematica [A] time = 0.23669, size = 183, normalized size = 0.87 \[ -\frac{\left (d+e x^2\right )^{q+1} \left (\left (d \sqrt{b^2-4 a c}+2 a e-b d\right ) \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )+\left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )\right )}{4 (q+1) \sqrt{b^2-4 a c} \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \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.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{q} x^{3}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q} x^{3}}{c x^{4} + b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{q} x^{3}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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