Optimal. Leaf size=125 \[ \frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d}-\frac {e \left (a+c x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{2 \left (c d^2+a e^2\right ) (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {771, 441, 440,
455, 70} \begin {gather*} \frac {x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d}-\frac {e \left (a+c x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (c x^2+a\right )}{c d^2+a e^2}\right )}{2 (p+1) \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 440
Rule 441
Rule 455
Rule 771
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^p}{d+e x} \, dx &=\int \left (\frac {d \left (a+c x^2\right )^p}{d^2-e^2 x^2}+\frac {e x \left (a+c x^2\right )^p}{-d^2+e^2 x^2}\right ) \, dx\\ &=d \int \frac {\left (a+c x^2\right )^p}{d^2-e^2 x^2} \, dx+e \int \frac {x \left (a+c x^2\right )^p}{-d^2+e^2 x^2} \, dx\\ &=\frac {1}{2} e \text {Subst}\left (\int \frac {(a+c x)^p}{-d^2+e^2 x} \, dx,x,x^2\right )+\left (d \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx\\ &=\frac {x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d}-\frac {e \left (a+c x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+c x^2\right )}{c d^2+a e^2}\right )}{2 \left (c d^2+a e^2\right ) (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.17, size = 131, normalized size = 1.05 \begin {gather*} \frac {\left (\frac {e \left (-\sqrt {-\frac {a}{c}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{d+e x}\right )^{-p} \left (a+c x^2\right )^p F_1\left (-2 p;-p,-p;1-2 p;\frac {d-\sqrt {-\frac {a}{c}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{c}} e}{d+e x}\right )}{2 e p} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+a \right )^{p}}{e x +d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + c x^{2}\right )^{p}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^p}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________