Optimal. Leaf size=92 \[ \frac {b d \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac {e \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {764, 266, 65, 365, 364} \[ \frac {b d \left (a+b x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac {e \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 364
Rule 365
Rule 764
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx &=d \int \frac {\left (a+b x^2\right )^p}{x^3} \, dx+e \int \frac {\left (a+b x^2\right )^p}{x^2} \, dx\\ &=\frac {1}{2} d \operatorname {Subst}\left (\int \frac {(a+b x)^p}{x^2} \, dx,x,x^2\right )+\left (e \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}{x^2} \, dx\\ &=-\frac {e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x}+\frac {b d \left (a+b x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a^2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 0.97 \[ \frac {1}{2} \left (a+b x^2\right )^p \left (\frac {b d \left (a+b x^2\right ) \, _2F_1\left (2,p+1;p+2;\frac {b x^2}{a}+1\right )}{a^2 (p+1)}-\frac {2 e \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right ) \left (b \,x^{2}+a \right )^{p}}{x^{3}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^p\,\left (d+e\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 13.92, size = 71, normalized size = 0.77 \[ - \frac {a^{p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{x} - \frac {b^{p} d x^{2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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