Optimal. Leaf size=168 \[ -\frac {d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac {e \left (a e^2+3 b d^2 (1+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{a}-\frac {d \left (3 a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a^2 (1+p)} \]
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Rubi [A]
time = 0.15, antiderivative size = 168, 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 = {1821, 778, 272,
67, 252, 251} \begin {gather*} -\frac {d \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 p\right ) \, _2F_1\left (1,p+1;p+2;\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac {d^3 \left (a+b x^2\right )^{p+1}}{2 a x^2}+\frac {e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a e^2+3 b d^2 (2 p+1)\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{a}-\frac {3 d^2 e \left (a+b x^2\right )^{p+1}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 251
Rule 252
Rule 272
Rule 778
Rule 1821
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x^3} \, dx &=-\frac {d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {\int \frac {\left (a+b x^2\right )^p \left (-6 a d^2 e-2 d \left (3 a e^2+b d^2 p\right ) x-2 a e^3 x^2\right )}{x^2} \, dx}{2 a}\\ &=-\frac {d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac {\int \frac {\left (2 a d \left (3 a e^2+b d^2 p\right )+2 a e \left (a e^2+3 b d^2 (1+2 p)\right ) x\right ) \left (a+b x^2\right )^p}{x} \, dx}{2 a^2}\\ &=-\frac {d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac {\left (d \left (3 a e^2+b d^2 p\right )\right ) \int \frac {\left (a+b x^2\right )^p}{x} \, dx}{a}+\frac {\left (e \left (a e^2+3 b d^2 (1+2 p)\right )\right ) \int \left (a+b x^2\right )^p \, dx}{a}\\ &=-\frac {d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac {\left (d \left (3 a e^2+b d^2 p\right )\right ) \text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^2\right )}{2 a}+\frac {\left (e \left (a e^2+3 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 \left (1+\frac {b x^2}{a}\right )^p \, dx}{a}\\ &=-\frac {d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac {3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac {e \left (a e^2+3 b d^2 (1+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 )}{a}-\frac {d \left (3 a e^2+b d^2 p\right ) \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a^2 (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 174, normalized size = 1.04 \begin {gather*} -\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (6 a^2 d^2 e (1+p) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )+x \left (-2 a^2 e^3 (1+p) x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )+d \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p \left (3 a e^2 \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )-b d^2 \, _2F_1\left (2,1+p;2+p;1+\frac {b x^2}{a}\right )\right )\right )\right )}{2 a^2 (1+p) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{3} \left (b \,x^{2}+a \right )^{p}}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [C] Result contains complex when optimal does not.
time = 8.80, size = 150, normalized size = 0.89 \begin {gather*} - \frac {3 a^{p} d^{2} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{x} + a^{p} e^{3} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} - \frac {b^{p} d^{3} x^{2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - p\right )} - \frac {3 b^{p} d e^{2} x^{2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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