Optimal. Leaf size=127 \[ -\frac {\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right ) \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac {d^2 \left (a+b x^2\right )^{p+1}}{2 a x^2}-\frac {2 d e \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x} \]
[Out]
________________________________________________________________________________________
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, 365, 364, 266, 65} \[ -\frac {\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right ) \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac {d^2 \left (a+b x^2\right )^{p+1}}{2 a x^2}-\frac {2 d e \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 266
Rule 364
Rule 365
Rule 764
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 \left (a+b x^2\right )^p}{x^3} \, dx &=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {\int \frac {\left (-4 a d e-2 \left (a e^2+b d^2 p\right ) x\right ) \left (a+b x^2\right )^p}{x^2} \, dx}{2 a}\\ &=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 a x^2}+(2 d e) \int \frac {\left (a+b x^2\right )^p}{x^2} \, dx+\frac {\left (a e^2+b d^2 p\right ) \int \frac {\left (a+b x^2\right )^p}{x} \, dx}{a}\\ &=-\frac {d^2 \left (a+b x^2\right )^{1+p}}{2 a x^2}+\frac {\left (a e^2+b d^2 p\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^2\right )}{2 a}+\left (2 d 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 {d^2 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {2 d 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 {\left (a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a^2 (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 119, normalized size = 0.94 \[ \frac {1}{2} \left (a+b x^2\right )^p \left (-\frac {\left (a+b x^2\right ) \left (a e^2 \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )-b d^2 \, _2F_1\left (2,p+1;p+2;\frac {b x^2}{a}+1\right )\right )}{a^2 (p+1)}-\frac {4 d e \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (b x^{2} + a\right )}^{p}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{2} \left (b \,x^{2}+a \right )^{p}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 18.62, size = 119, normalized size = 0.94 \[ - \frac {2 a^{p} d e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{x} - \frac {b^{p} d^{2} x^{2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} - \frac {b^{p} e^{2} x^{2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________