Optimal. Leaf size=244 \[ -\frac{2 b p x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,1;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{a e^2+b d^2}+\frac{b (2 p+1) x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{a e^2+b d^2}-\frac{b d e \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{(p+1) \left (a e^2+b d^2\right )^2}+\frac{e^2 x \left (a+b x^2\right )^{p+1}}{\left (d^2-e^2 x^2\right ) \left (a e^2+b d^2\right )} \]
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Rubi [A] time = 0.179792, antiderivative size = 191, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {757, 430, 429, 444, 68, 511, 510} \[ \frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2}+\frac{e^2 x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} F_1\left (\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d^4}-\frac{b d e \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{(p+1) \left (a e^2+b d^2\right )^2} \]
Warning: Unable to verify antiderivative.
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Rule 757
Rule 430
Rule 429
Rule 444
Rule 68
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^p}{(d+e x)^2} \, dx &=\int \left (\frac{d^2 \left (a+b x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}-\frac{2 d e x \left (a+b x^2\right )^p}{\left (d^2-e^2 x^2\right )^2}+\frac{e^2 x^2 \left (a+b x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2}\right ) \, dx\\ &=d^2 \int \frac{\left (a+b x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx-(2 d e) \int \frac{x \left (a+b x^2\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+e^2 \int \frac{x^2 \left (a+b x^2\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx\\ &=-\left ((d e) \operatorname{Subst}\left (\int \frac{(a+b x)^p}{\left (d^2-e^2 x\right )^2} \, dx,x,x^2\right )\right )+\left (d^2 \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{b x^2}{a}\right )^p}{\left (d^2-e^2 x^2\right )^2} \, dx+\left (e^2 \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int \frac{x^2 \left (1+\frac{b x^2}{a}\right )^p}{\left (-d^2+e^2 x^2\right )^2} \, dx\\ &=\frac{x \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} F_1\left (\frac{1}{2};-p,2;\frac{3}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{d^2}+\frac{e^2 x^3 \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} F_1\left (\frac{3}{2};-p,2;\frac{5}{2};-\frac{b x^2}{a},\frac{e^2 x^2}{d^2}\right )}{3 d^4}-\frac{b d e \left (a+b x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;\frac{e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{\left (b d^2+a e^2\right )^2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0874415, size = 141, normalized size = 0.58 \[ \frac{\left (a+b x^2\right )^p \left (\frac{e \left (x-\sqrt{-\frac{a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d-\sqrt{-\frac{a}{b}} e}{d+e x},\frac{d+\sqrt{-\frac{a}{b}} e}{d+e x}\right )}{e (2 p-1) (d+e x)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.658, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, 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 (b x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{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 (b x^{2} + a\right )}^{p}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, 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 (b x^{2} + a\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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