Optimal. Leaf size=347 \[ -\frac {e (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (3+2 p)}-\frac {c d e (2+p) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (1+p) (3+2 p)}+\frac {c \left (a e^2-c d^2 (3+2 p)\right ) \left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right )^2 (1+2 p) (3+2 p)} \]
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Rubi [A]
time = 0.12, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {759, 821, 741}
\begin {gather*} \frac {c \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (a e^2-c d^2 (2 p+3)\right ) \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) (2 p+3) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )^2}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(2 p+3) \left (a e^2+c d^2\right )}-\frac {c d e (p+2) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{(p+1) (2 p+3) \left (a e^2+c d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 741
Rule 759
Rule 821
Rubi steps
\begin {align*} \int (d+e x)^{-4-2 p} \left (a+c x^2\right )^p \, dx &=-\frac {e (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (3+2 p)}-\frac {c \int (d+e x)^{-3-2 p} (-d (3+2 p)+e x) \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right ) (3+2 p)}\\ &=-\frac {e (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (3+2 p)}-\frac {c d e (2+p) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (1+p) (3+2 p)}-\frac {\left (c \left (a e^2-c d^2 (3+2 p)\right )\right ) \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right )^2 (3+2 p)}\\ &=-\frac {e (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (3+2 p)}-\frac {c d e (2+p) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (1+p) (3+2 p)}+\frac {c \left (a e^2-c d^2 (3+2 p)\right ) \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 )^2 (1+2 p) (3+2 p)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1439\) vs. \(2(347)=694\).
time = 12.45, size = 1439, normalized size = 4.15 \begin {gather*} -\frac {2 \left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}\right )^{-p} (d+e x)^{-3-2 p} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )^{1+p} \Gamma (-2 (1+p)) \left (\left (d+\sqrt {-\frac {a}{c}} e\right )^3 \left (\sqrt {-\frac {a}{c}}+x\right ) \Gamma (1-2 p) \Gamma (-p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+3 \left (d+\sqrt {-\frac {a}{c}} e\right )^3 p \left (\sqrt {-\frac {a}{c}}+x\right ) \Gamma (1-2 p) \Gamma (-p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+2 \left (d+\sqrt {-\frac {a}{c}} e\right )^3 p^2 \left (\sqrt {-\frac {a}{c}}+x\right ) \Gamma (1-2 p) \Gamma (-p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+\left (d+\sqrt {-\frac {a}{c}} e\right )^2 \left (\sqrt {-\frac {a}{c}}+x\right ) (d+e x) \Gamma (1-2 p) \Gamma (-p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+2 \left (d+\sqrt {-\frac {a}{c}} e\right )^2 p \left (\sqrt {-\frac {a}{c}}+x\right ) (d+e x) \Gamma (1-2 p) \Gamma (-p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (\sqrt {-\frac {a}{c}}+x\right ) (d+e x)^2 \Gamma (1-2 p) \Gamma (-p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )-3 \sqrt {-\frac {a}{c}} \left (d+\sqrt {-\frac {a}{c}} e\right )^2 (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )-4 \sqrt {-\frac {a}{c}} \left (d+\sqrt {-\frac {a}{c}} e\right )^2 p (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+4 \sqrt {-\frac {a}{c}} \left (d+\sqrt {-\frac {a}{c}} e\right ) p (d+e x)^2 \Gamma (1-p) \Gamma (-2 p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+3 \sqrt {-\frac {a}{c}} (d+e x)^3 \Gamma (1-p) \Gamma (-2 p) \, _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 (\sqrt {-\frac {a}{c}}+x\right )}\right )+\sqrt {-\frac {a}{c}} \left (d+\sqrt {-\frac {a}{c}} e\right )^2 (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _3F_2\left (2,2,1-p;1,1-2 p;\frac {2 \sqrt {-\frac {a}{c}} (d+e x)}{\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (\sqrt {-\frac {a}{c}}+x\right )}\right )-2 \sqrt {-\frac {a}{c}} \left (d+\sqrt {-\frac {a}{c}} e\right ) (d+e x)^2 \Gamma (1-p) \Gamma (-2 p) \, _3F_2\left (2,2,1-p;1,1-2 p;\frac {2 \sqrt {-\frac {a}{c}} (d+e x)}{\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (\sqrt {-\frac {a}{c}}+x\right )}\right )+\sqrt {-\frac {a}{c}} (d+e x)^3 \Gamma (1-p) \Gamma (-2 p) \, _3F_2\left (2,2,1-p;1,1-2 p;\frac {2 \sqrt {-\frac {a}{c}} (d+e x)}{\left (d+\sqrt {-\frac {a}{c}} e\right ) \left (\sqrt {-\frac {a}{c}}+x\right )}\right )\right )}{e \left (d+\sqrt {-\frac {a}{c}} e\right )^3 (3+2 p) \left (\sqrt {-\frac {a}{c}}+x\right ) \Gamma (1-2 p) \Gamma (-2 p) \Gamma (-p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{-4-2 p} \left (c \,x^{2}+a \right )^{p}\, dx\]
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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