Optimal. Leaf size=199 \[ \frac {x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d}+\frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^2 e^3 (p+1)}+\frac {\left (a+b x^2\right )^{p+2}}{2 b^2 e (p+2)}-\frac {d^4 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^3 (p+1) \left (a e^2+b d^2\right )} \]
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Rubi [A] time = 0.23, 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 = {959, 511, 510, 446, 88, 68} \[ \frac {x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} F_1\left (\frac {5}{2};-p,1;\frac {7}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{5 d}+\frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^2 e^3 (p+1)}+\frac {\left (a+b x^2\right )^{p+2}}{2 b^2 e (p+2)}-\frac {d^4 \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^3 (p+1) \left (a e^2+b d^2\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 88
Rule 446
Rule 510
Rule 511
Rule 959
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 \operatorname {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 \operatorname {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 \operatorname {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*}
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Mathematica [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {x^4 \left (a+b x^2\right )^p}{d+e x} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p} x^{4}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x^{4}}{e x + d}\,{d x} \]
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 \[ \int \frac {x^{4} \left (b \,x^{2}+a \right )^{p}}{e x +d}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x^{4}}{e x + d}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (b\,x^2+a\right )}^p}{d+e\,x} \,d x \]
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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