Optimal. Leaf size=189 \[ \frac {x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} F_1\left (\frac {1}{4};-p,2;\frac {5}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c^2}+\frac {e^2 x^5 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} F_1\left (\frac {5}{4};-p,2;\frac {9}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{5 c^4}-\frac {2 e x^3 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} F_1\left (\frac {3}{4};-p,2;\frac {7}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1240, 430, 429, 511, 510} \[ \frac {x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} F_1\left (\frac {1}{4};-p,2;\frac {5}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c^2}-\frac {2 e x^3 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} F_1\left (\frac {3}{4};-p,2;\frac {7}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^3}+\frac {e^2 x^5 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} F_1\left (\frac {5}{4};-p,2;\frac {9}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{5 c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 429
Rule 430
Rule 510
Rule 511
Rule 1240
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx &=\int \left (\frac {c^2 \left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2}-\frac {2 c e x^2 \left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2}+\frac {e^2 x^4 \left (a+b x^4\right )^p}{\left (-c^2+e^2 x^4\right )^2}\right ) \, dx\\ &=c^2 \int \frac {\left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx-(2 c e) \int \frac {x^2 \left (a+b x^4\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx+e^2 \int \frac {x^4 \left (a+b x^4\right )^p}{\left (-c^2+e^2 x^4\right )^2} \, dx\\ &=\left (c^2 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^4}{a}\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx-\left (2 c e \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {x^2 \left (1+\frac {b x^4}{a}\right )^p}{\left (c^2-e^2 x^4\right )^2} \, dx+\left (e^2 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \frac {x^4 \left (1+\frac {b x^4}{a}\right )^p}{\left (-c^2+e^2 x^4\right )^2} \, dx\\ &=\frac {x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {1}{4};-p,2;\frac {5}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{c^2}-\frac {2 e x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {3}{4};-p,2;\frac {7}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{3 c^3}+\frac {e^2 x^5 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} F_1\left (\frac {5}{4};-p,2;\frac {9}{4};-\frac {b x^4}{a},\frac {e^2 x^4}{c^2}\right )}{5 c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b x^4\right )^p}{\left (c+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{p}}{e^{2} x^{4} + 2 \, c e x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{p}}{\left (e \,x^{2}+c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{p}}{{\left (e x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^4+a\right )}^p}{{\left (e\,x^2+c\right )}^2} \,d x \]
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]
________________________________________________________________________________________