Optimal. Leaf size=127 \[ \frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{a}-\frac {d^2 \left (a+b x^2\right )^{p+1}}{a x}-\frac {d e \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )}{a (p+1)} \]
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Rubi [A] time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1807, 764, 266, 65, 246, 245} \[ \frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{a}-\frac {d^2 \left (a+b x^2\right )^{p+1}}{a x}-\frac {d e \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )}{a (p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 245
Rule 246
Rule 266
Rule 764
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b x^2\right )^p}{x^2} \, dx &=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{a x}-\frac {\int \frac {\left (-2 a d e-\left (a e^2+b d^2 (1+2 p)\right ) x\right ) \left (a+b x^2\right )^p}{x} \, dx}{a}\\ &=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{a x}+(2 d e) \int \frac {\left (a+b x^2\right )^p}{x} \, dx+\frac {\left (a e^2+b d^2 (1+2 p)\right ) \int \left (a+b x^2\right )^p \, dx}{a}\\ &=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{a x}+(d e) \operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^2\right )+\frac {\left (\left (a e^2+b d^2 (1+2 p)\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx}{a}\\ &=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{a x}+\frac {\left (a e^2+b d^2 (1+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{a}-\frac {d e \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )}{a (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 134, normalized size = 1.06 \[ -\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a d^2 (p+1) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )+e x \left (d \left (a+b x^2\right ) \left (\frac {b x^2}{a}+1\right )^p \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )-a e (p+1) x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )\right )\right )}{a (p+1) x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (b x^{2} + a\right )}^{p}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{2} \left (b \,x^{2}+a \right )^{p}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.57, size = 95, normalized size = 0.75 \[ - \frac {a^{p} d^{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{x} + a^{p} e^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} - \frac {b^{p} d e x^{2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{\Gamma \left (1 - p\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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