Optimal. Leaf size=336 \[ -\frac {b p x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+1)\right ) F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{e \left (a e^2+b d^2\right )^2}+\frac {b (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a e^2+b d^2 p\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e \left (a e^2+b d^2\right )^2}+\frac {b d p \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 (p+1) \left (a e^2+b d^2\right )^3}-\frac {\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right )}{(d+e x) \left (a e^2+b d^2\right )^2}+\frac {d \left (a+b x^2\right )^{p+1}}{2 (d+e x)^2 \left (a e^2+b d^2\right )} \]
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Rubi [A] time = 0.40, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {835, 844, 246, 245, 757, 430, 429, 444, 68} \[ -\frac {b p x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+1)\right ) F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b x^2}{a},\frac {e^2 x^2}{d^2}\right )}{e \left (a e^2+b d^2\right )^2}+\frac {b (2 p+1) x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a e^2+b d^2 p\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{e \left (a e^2+b d^2\right )^2}+\frac {b d p \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (1,p+1;p+2;\frac {e^2 \left (b x^2+a\right )}{b d^2+a e^2}\right )}{2 (p+1) \left (a e^2+b d^2\right )^3}-\frac {\left (a+b x^2\right )^{p+1} \left (a e^2+b d^2 p\right )}{(d+e x) \left (a e^2+b d^2\right )^2}+\frac {d \left (a+b x^2\right )^{p+1}}{2 (d+e x)^2 \left (a e^2+b d^2\right )} \]
Antiderivative was successfully verified.
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Rule 68
Rule 245
Rule 246
Rule 429
Rule 430
Rule 444
Rule 757
Rule 835
Rule 844
Rubi steps
\begin {align*} \int \frac {x \left (a+b x^2\right )^p}{(d+e x)^3} \, dx &=\frac {d \left (a+b x^2\right )^{1+p}}{2 \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\int \frac {(-2 a e+2 b d p x) \left (a+b x^2\right )^p}{(d+e x)^2} \, dx}{2 \left (b d^2+a e^2\right )}\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p}}{\left (b d^2+a e^2\right )^2 (d+e x)}+\frac {\int \frac {\left (2 a b d e (1-p)+2 b (1+2 p) \left (a e^2+b d^2 p\right ) x\right ) \left (a+b x^2\right )^p}{d+e x} \, dx}{2 \left (b d^2+a e^2\right )^2}\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p}}{\left (b d^2+a e^2\right )^2 (d+e x)}+\frac {\left (b (1+2 p) \left (a e^2+b d^2 p\right )\right ) \int \left (a+b x^2\right )^p \, dx}{e \left (b d^2+a e^2\right )^2}-\frac {\left (b d p \left (3 a e^2+b d^2 (1+2 p)\right )\right ) \int \frac {\left (a+b x^2\right )^p}{d+e x} \, dx}{e \left (b d^2+a e^2\right )^2}\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p}}{\left (b d^2+a e^2\right )^2 (d+e x)}-\frac {\left (b d p \left (3 a e^2+b d^2 (1+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 \left (b d^2+a e^2\right )^2}+\frac {\left (b (1+2 p) \left (a e^2+b d^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 \left (b d^2+a e^2\right )^2}\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p}}{\left (b d^2+a e^2\right )^2 (d+e x)}+\frac {b (1+2 p) \left (a e^2+b d^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 \left (b d^2+a e^2\right )^2}-\frac {\left (b d p \left (3 a e^2+b d^2 (1+2 p)\right )\right ) \int \frac {x \left (a+b x^2\right )^p}{-d^2+e^2 x^2} \, dx}{\left (b d^2+a e^2\right )^2}-\frac {\left (b d^2 p \left (3 a e^2+b d^2 (1+2 p)\right )\right ) \int \frac {\left (a+b x^2\right )^p}{d^2-e^2 x^2} \, dx}{e \left (b d^2+a e^2\right )^2}\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p}}{\left (b d^2+a e^2\right )^2 (d+e x)}+\frac {b (1+2 p) \left (a e^2+b d^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 \left (b d^2+a e^2\right )^2}-\frac {\left (b d p \left (3 a e^2+b d^2 (1+2 p)\right )\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^p}{-d^2+e^2 x} \, dx,x,x^2\right )}{2 \left (b d^2+a e^2\right )^2}-\frac {\left (b d^2 p \left (3 a e^2+b d^2 (1+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 \left (b d^2+a e^2\right )^2}\\ &=\frac {d \left (a+b x^2\right )^{1+p}}{2 \left (b d^2+a e^2\right ) (d+e x)^2}-\frac {\left (a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p}}{\left (b d^2+a e^2\right )^2 (d+e x)}-\frac {b p \left (3 a e^2+b d^2 (1+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 \left (b d^2+a e^2\right )^2}+\frac {b (1+2 p) \left (a e^2+b d^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 \left (b d^2+a e^2\right )^2}+\frac {b d p \left (3 a e^2+b d^2 (1+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 \left (b d^2+a e^2\right )^3 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 229, normalized size = 0.68 \[ \frac {\left (a+b x^2\right )^p \left (\frac {e \left (x-\sqrt {-\frac {a}{b}}\right )}{d+e x}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{b}}+x\right )}{d+e x}\right )^{-p} \left (2 (p-1) (d+e x) F_1\left (1-2 p;-p,-p;2-2 p;\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )+d (1-2 p) F_1\left (2-2 p;-p,-p;3-2 p;\frac {d-\sqrt {-\frac {a}{b}} e}{d+e x},\frac {d+\sqrt {-\frac {a}{b}} e}{d+e x}\right )\right )}{2 e^2 (p-1) (2 p-1) (d+e x)^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{p} x}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x \left (b \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{p} x}{{\left (e x + d\right )}^{3}}\,{d x} \]
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 \[ \int \frac {x\,{\left (b\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^3} \,d x \]
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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