Optimal. Leaf size=92 \[ -\frac {e \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {c x^2}{a}\right )}{x}+\frac {c d \left (a+c x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1+\frac {c x^2}{a}\right )}{2 a^2 (1+p)} \]
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Rubi [A]
time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {778, 272, 67,
372, 371} \begin {gather*} \frac {c d \left (a+c x^2\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {c x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac {e \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {c x^2}{a}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rule 371
Rule 372
Rule 778
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (a+c x^2\right )^p}{x^3} \, dx &=d \int \frac {\left (a+c x^2\right )^p}{x^3} \, dx+e \int \frac {\left (a+c x^2\right )^p}{x^2} \, dx\\ &=\frac {1}{2} d \text {Subst}\left (\int \frac {(a+c x)^p}{x^2} \, dx,x,x^2\right )+\left (e \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^2}{a}\right )^p}{x^2} \, dx\\ &=-\frac {e \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {c x^2}{a}\right )}{x}+\frac {c d \left (a+c x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1+\frac {c x^2}{a}\right )}{2 a^2 (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 89, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (a+c x^2\right )^p \left (-\frac {2 e \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {c x^2}{a}\right )}{x}+\frac {c d \left (a+c x^2\right ) \, _2F_1\left (2,1+p;2+p;1+\frac {c x^2}{a}\right )}{a^2 (1+p)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right ) \left (c \,x^{2}+a \right )^{p}}{x^{3}}\, dx\]
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 = 6.71, size = 71, normalized size = 0.77 \begin {gather*} - \frac {a^{p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{x} - \frac {c^{p} d 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 }}{c x^{2}}} \right )}}{2 x^{2} \Gamma \left (2 - 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 (c\,x^2+a\right )}^p\,\left (d+e\,x\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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