Optimal. Leaf size=559 \[ -\frac {c^2 \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )^4}+\frac {c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac {c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac {c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.73, antiderivative size = 559, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {745, 837, 807, 727} \[ -\frac {c^2 \left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )^4}+\frac {c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac {c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac {c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 727
Rule 745
Rule 807
Rule 837
Rubi steps
\begin {align*} \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx &=-\frac {e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}-\frac {c \int (d+e x)^{-5-2 p} (-d (5+2 p)+3 e x) \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right ) (5+2 p)}\\ &=-\frac {e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}-\frac {c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}+\frac {c \int (d+e x)^{-4-2 p} \left (-2 (2+p) \left (3 a e^2-c d^2 (5+2 p)\right )-4 c d e (4+p) x\right ) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}\\ &=-\frac {e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac {c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (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) (3+2 p) (5+2 p)}-\frac {c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}-\frac {c \int (d+e x)^{-3-2 p} \left (2 c d (3+2 p) \left (a e^2 (14+5 p)-c d^2 \left (10+9 p+2 p^2\right )\right )-2 c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) x\right ) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}\\ &=-\frac {e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac {c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (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) (3+2 p) (5+2 p)}+\frac {c^2 d e (3+p) \left (a e^2 (8+5 p)-c d^2 \left (8+7 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac {c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}+\frac {\left (c^2 \left (3 a^2 e^4-6 a c d^2 e^2 (5+2 p)+c^2 d^4 \left (15+16 p+4 p^2\right )\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 )^4 (3+2 p) (5+2 p)}\\ &=-\frac {e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac {c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (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) (3+2 p) (5+2 p)}+\frac {c^2 d e (3+p) \left (a e^2 (8+5 p)-c d^2 \left (8+7 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac {c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}-\frac {c^2 \left (3 a^2 e^4-6 a c d^2 e^2 (5+2 p)+c^2 d^4 \left (15+16 p+4 p^2\right )\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 )^4 (1+2 p) (3+2 p) (5+2 p)}\\ \end {align*}
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Mathematica [B] time = 36.29, size = 5685, normalized size = 10.17 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6}, x\right ) \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+a \right )^{p} \left (e x +d \right )^{-2 p -6}\, dx \]
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 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+6}} \,d x \]
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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