Optimal. Leaf size=321 \[ \frac {d x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+3)\right ) F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{e^3 \left (a e^2+b d^2\right )}-\frac {d^2 \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (2 p+3)\right ) \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )^2}-\frac {d x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a e^2+b d^2 (2 p+3)\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^3 \left (a e^2+b d^2\right )}+\frac {d^3 \left (a+b x^2\right )^{p+1}}{e^2 (d+e x) \left (a e^2+b d^2\right )}+\frac {\left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \]
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Rubi [A] time = 0.55, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1651, 1654, 844, 246, 245, 757, 430, 429, 444, 68} \[ \frac {d x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+3)\right ) F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{e^3 \left (a e^2+b d^2\right )}-\frac {d x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (2 a e^2+b d^2 (2 p+3)\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^3 \left (a e^2+b d^2\right )}-\frac {d^2 \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (2 p+3)\right ) \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 e^2 (p+1) \left (a e^2+b d^2\right )^2}+\frac {d^3 \left (a+b x^2\right )^{p+1}}{e^2 (d+e x) \left (a e^2+b d^2\right )}+\frac {\left (a+b x^2\right )^{p+1}}{2 b e^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 245
Rule 246
Rule 429
Rule 430
Rule 444
Rule 757
Rule 844
Rule 1651
Rule 1654
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b x^2\right )^p}{(d+e x)^2} \, dx &=\frac {d^3 \left (a+b x^2\right )^{1+p}}{e^2 \left (b d^2+a e^2\right ) (d+e x)}-\frac {\int \frac {\left (a+b x^2\right )^p \left (-\frac {a d^2}{e}+d \left (a+\frac {2 b d^2 (1+p)}{e^2}\right ) x-\frac {\left (b d^2+a e^2\right ) x^2}{e}\right )}{d+e x} \, dx}{b d^2+a e^2}\\ &=\frac {\left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}+\frac {d^3 \left (a+b x^2\right )^{1+p}}{e^2 \left (b d^2+a e^2\right ) (d+e x)}-\frac {\int \frac {\left (-2 a b d^2 e (1+p)+2 b d (1+p) \left (2 a e^2+b d^2 (3+2 p)\right ) x\right ) \left (a+b x^2\right )^p}{d+e x} \, dx}{2 b e^2 \left (b d^2+a e^2\right ) (1+p)}\\ &=\frac {\left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}+\frac {d^3 \left (a+b x^2\right )^{1+p}}{e^2 \left (b d^2+a e^2\right ) (d+e x)}-\frac {\left (d \left (2 a e^2+b d^2 (3+2 p)\right )\right ) \int \left (a+b x^2\right )^p \, dx}{e^3 \left (b d^2+a e^2\right )}+\frac {\left (d^2 \left (3 a e^2+b d^2 (3+2 p)\right )\right ) \int \frac {\left (a+b x^2\right )^p}{d+e x} \, dx}{e^3 \left (b d^2+a e^2\right )}\\ &=\frac {\left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}+\frac {d^3 \left (a+b x^2\right )^{1+p}}{e^2 \left (b d^2+a e^2\right ) (d+e x)}+\frac {\left (d^2 \left (3 a e^2+b d^2 (3+2 p)\right )\right ) \int \left (\frac {d \left (a+b x^2\right )^p}{d^2-e^2 x^2}+\frac {e x \left (a+b x^2\right )^p}{-d^2+e^2 x^2}\right ) \, dx}{e^3 \left (b d^2+a e^2\right )}-\frac {\left (d \left (2 a e^2+b d^2 (3+2 p)\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx}{e^3 \left (b d^2+a e^2\right )}\\ &=\frac {\left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}+\frac {d^3 \left (a+b x^2\right )^{1+p}}{e^2 \left (b d^2+a e^2\right ) (d+e x)}-\frac {d \left (2 a e^2+b d^2 (3+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^3 \left (b d^2+a e^2\right )}+\frac {\left (d^3 \left (3 a e^2+b d^2 (3+2 p)\right )\right ) \int \frac {\left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx}{e^3 \left (b d^2+a e^2\right )}+\frac {\left (d^2 \left (3 a e^2+b d^2 (3+2 p)\right )\right ) \int \frac {x \left (a+b x^2\right )^p}{-d^2+e^2 x^2} \, dx}{e^2 \left (b d^2+a e^2\right )}\\ &=\frac {\left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}+\frac {d^3 \left (a+b x^2\right )^{1+p}}{e^2 \left (b d^2+a e^2\right ) (d+e x)}-\frac {d \left (2 a e^2+b d^2 (3+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^3 \left (b d^2+a e^2\right )}+\frac {\left (d^2 \left (3 a e^2+b d^2 (3+2 p)\right )\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^p}{-d^2+e^2 x} \, dx,x,x^2\right )}{2 e^2 \left (b d^2+a e^2\right )}+\frac {\left (d^3 \left (3 a e^2+b d^2 (3+2 p)\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^p}{d^2-e^2 x^2} \, dx}{e^3 \left (b d^2+a e^2\right )}\\ &=\frac {\left (a+b x^2\right )^{1+p}}{2 b e^2 (1+p)}+\frac {d^3 \left (a+b x^2\right )^{1+p}}{e^2 \left (b d^2+a e^2\right ) (d+e x)}+\frac {d \left (3 a e^2+b d^2 (3+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{e^3 \left (b d^2+a e^2\right )}-\frac {d \left (2 a e^2+b d^2 (3+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e^3 \left (b d^2+a e^2\right )}-\frac {d^2 \left (3 a e^2+b d^2 (3+2 p)\right ) \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 )^2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 343, normalized size = 1.07 \[ \frac {\left (a+b x^2\right )^p \left (-\frac {2 d^3 \left (\frac {e \left (x-\sqrt {-\frac {a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{(2 p-1) (d+e x)}+\frac {3 d^2 \left (\frac {e \left (x-\sqrt {-\frac {a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )}{p}+e \left (\frac {e \left (-a \left (\frac {b x^2}{a}+1\right )^{-p}+a+b x^2\right )}{b p+b}-4 d x \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )\right )\right )}{2 e^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (b \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\left (b\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]
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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