Optimal. Leaf size=163 \[ -\frac {d \left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}-\frac {e x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d^2}+\frac {d^3 \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {e^2 \left (a+b x^2\right )}{b d^2+a e^2}\right )}{2 e^2 \left (b d^2+a e^2\right ) (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 163, 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 = {973, 457, 81,
70, 525, 524} \begin {gather*} -\frac {e 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^2}+\frac {d^3 \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^2 (p+1) \left (a e^2+b d^2\right )}-\frac {d \left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 81
Rule 457
Rule 524
Rule 525
Rule 973
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b x^2\right )^p}{d+e x} \, dx &=d \int \frac {x^3 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx-e \int \frac {x^4 \left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx\\ &=\frac {1}{2} d \text {Subst}\left (\int \frac {x (a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )-\left (e \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 {d \left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}-\frac {e 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^2}+\frac {d^3 \text {Subst}\left (\int \frac {(a+b x)^p}{d^2-e^2 x} \, dx,x,x^2\right )}{2 e^2}\\ &=-\frac {d \left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}-\frac {e 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^2}+\frac {d^3 \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^2 \left (b d^2+a e^2\right ) (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.34, size = 260, normalized size = 1.60 \begin {gather*} \frac {\left (a+b x^2\right )^p \left (-\frac {3 d^3 \left (\frac {e \left (-\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{p}+\frac {e \left (1+\frac {b x^2}{a}\right )^{-p} \left (6 b d^2 (1+p) x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )+e \left (-3 d \left (b x^2 \left (1+\frac {b x^2}{a}\right )^p+a \left (-1+\left (1+\frac {b x^2}{a}\right )^p\right )\right )+2 b e (1+p) x^3 \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right )\right )\right )}{b (1+p)}\right )}{6 e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (b \,x^{2}+a \right )^{p}}{e x +d}\, dx\]
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^3\,{\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]
________________________________________________________________________________________