Optimal. Leaf size=199 \[ \frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 e^3 (1+p)}+\frac {\left (a+b x^2\right )^{2+p}}{2 b^2 e (2+p)}+\frac {x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d}-\frac {d^4 \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e^3 \left (b d^2+a e^2\right ) (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {973, 525, 524,
457, 90, 70} \begin {gather*} \frac {x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d}+\frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^2 e^3 (p+1)}+\frac {\left (a+b x^2\right )^{p+2}}{2 b^2 e (p+2)}-\frac {d^4 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^3 (p+1) \left (a e^2+b d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 90
Rule 457
Rule 524
Rule 525
Rule 973
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )^p}{d+e x} \, dx &=d \int \frac {x^4 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx-e \int \frac {x^5 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx\\ &=-\left (\frac {1}{2} e \text {Subst}\left (\int \frac {x^2 (a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )\right )+\left (d \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {x^4 \left (1+\frac {b x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx\\ &=\frac {x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d}-\frac {1}{2} e \text {Subst}\left (\int \left (\frac {\left (-b d^2+a e^2\right ) (a+b x)^p}{b e^4}-\frac {(a+b x)^{1+p}}{b e^2}+\frac {d^4 (a+b x)^p}{e^4 \left (d^2-e^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 e^3 (1+p)}+\frac {\left (a+b x^2\right )^{2+p}}{2 b^2 e (2+p)}+\frac {x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d}-\frac {d^4 \text {Subst}\left (\int \frac {(a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )}{2 e^3}\\ &=\frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 e^3 (1+p)}+\frac {\left (a+b x^2\right )^{2+p}}{2 b^2 e (2+p)}+\frac {x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d}-\frac {d^4 \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e^3 \left (b d^2+a e^2\right ) (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 0.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (a+b x^2\right )^p}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (b \,x^{2}+a \right )^{p}}{e x +d}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {x^4\,{\left (b\,x^2+a\right )}^p}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________