Optimal. Leaf size=270 \[ -\frac {c d \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {731, 727} \[ -\frac {c d \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 727
Rule 731
Rubi steps
\begin {align*} \int (d+e x)^{-3-2 p} \left (a+c x^2\right )^p \, dx &=-\frac {e (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (1+p)}+\frac {(c d) \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx}{c d^2+a e^2}\\ &=-\frac {e (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 \left (c d^2+a e^2\right ) (1+p)}-\frac {c d \left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 37.08, size = 368, normalized size = 1.36 \[ \frac {2^{-2 p-3} \Gamma \left (-p-\frac {1}{2}\right ) \left (a+c x^2\right )^p (d+e x)^{-2 (p+1)} \left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{e \sqrt {-\frac {a}{c}}+d}\right )^{-p} \left (1-\frac {d+e x}{e \sqrt {-\frac {a}{c}}+d}\right )^{p+1} \left (\Gamma (1-2 p) \Gamma (-p) \left (e \sqrt {-\frac {a}{c}}+d\right ) \left (e \left (2 p \sqrt {-\frac {a}{c}}+\sqrt {-\frac {a}{c}}+x\right )+2 d (p+1)\right ) \, _2F_1\left (1,-p;-2 p;\frac {2 \sqrt {-\frac {a}{c}} (d+e x)}{\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (x+\sqrt {-\frac {a}{c}}\right )}\right )+\frac {2 e \Gamma (1-p) \Gamma (-2 p) \left (c x \sqrt {-\frac {a}{c}}+a\right ) (d+e x) \, _2F_1\left (2,1-p;1-2 p;\frac {2 \sqrt {-\frac {a}{c}} (d+e x)}{\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (x+\sqrt {-\frac {a}{c}}\right )}\right )}{c \left (\sqrt {-\frac {a}{c}}+x\right )}\right )}{\sqrt {\pi } e (p+1) \Gamma (1-2 p) \Gamma (-2 p) \left (e \sqrt {-\frac {a}{c}}+d\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}, x\right ) \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+a \right )^{p} \left (e x +d \right )^{-2 p -3}\, dx \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+3}} \,d x \]
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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