Optimal. Leaf size=79 \[ x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} F_1\left (\frac {1}{2};-p,-q;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {430, 429} \[ x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} F_1\left (\frac {1}{2};-p,-q;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 429
Rule 430
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx &=\left (\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 \left (c+d x^2\right )^q \, dx\\ &=\left (\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \left (1+\frac {d x^2}{c}\right )^q \, dx\\ &=x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} F_1\left (\frac {1}{2};-p,-q;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.22, size = 172, normalized size = 2.18 \[ \frac {3 a c x \left (a+b x^2\right )^p \left (c+d x^2\right )^q F_1\left (\frac {1}{2};-p,-q;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{2 x^2 \left (b c p F_1\left (\frac {3}{2};1-p,-q;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+a d q F_1\left (\frac {3}{2};-p,1-q;\frac {5}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )+3 a c F_1\left (\frac {1}{2};-p,-q;\frac {3}{2};-\frac {b x^2}{a},-\frac {d x^2}{c}\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^q \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________