Optimal. Leaf size=163 \[ -\frac {e x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d^2}+\frac {d^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )}-\frac {d \left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \]
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Rubi [A] time = 0.15, 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 = {959, 446, 80, 68, 511, 510} \[ -\frac {e x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d^2}+\frac {d^3 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )}-\frac {d \left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 80
Rule 446
Rule 510
Rule 511
Rule 959
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 \operatorname {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 \operatorname {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*}
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Mathematica [A] time = 0.33, size = 260, normalized size = 1.60 \[ \frac {\left (a+b x^2\right )^p \left (\frac {e \left (\frac {b x^2}{a}+1\right )^{-p} \left (6 b d^2 (p+1) x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )+e \left (2 b e (p+1) x^3 \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right )-3 d \left (b x^2 \left (\frac {b x^2}{a}+1\right )^p+a \left (\left (\frac {b x^2}{a}+1\right )^p-1\right )\right )\right )\right )}{b (p+1)}-\frac {3 d^3 \left (\frac {e \left (x-\sqrt {-\frac {a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{p}\right )}{6 e^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, 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.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,{\left (b\,x^2+a\right )}^p}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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