Optimal. Leaf size=498 \[ -\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (5 a e^2+8 c d^2\right )+d \left (29 a e^2+32 c d^2\right )\right )}{21 e^5 \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (29 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (5 a e^2+11 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.454597, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {733, 811, 813, 844, 719, 424, 419} \[ \frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (5 a e^2+8 c d^2\right )+d \left (29 a e^2+32 c d^2\right )\right )}{21 e^5 \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (29 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (5 a e^2+11 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 733
Rule 811
Rule 813
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx &=-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{(10 c) \int \frac{x \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{(4 c) \int \frac{\left (3 a c d e-c \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{(d+e x)^{3/2}} \, dx}{7 e^3 \left (c d^2+a e^2\right )}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{(8 c) \int \frac{a c e \left (8 c d^2+5 a e^2\right )-c^2 d \left (32 c d^2+29 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{21 e^5 \left (c d^2+a e^2\right )}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{\left (8 c^2 \left (32 c d^2+5 a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{21 e^6}-\frac{\left (8 c^3 d \left (32 c d^2+29 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{21 e^6 \left (c d^2+a e^2\right )}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{\left (16 a c^{5/2} d \left (32 c d^2+29 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{21 \sqrt{-a} e^6 \left (c d^2+a e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (16 a c^{3/2} \left (32 c d^2+5 a e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{21 \sqrt{-a} e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{16 \sqrt{-a} c^{5/2} d \left (32 c d^2+29 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \left (c d^2+a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{16 \sqrt{-a} c^{3/2} \left (32 c d^2+5 a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 3.99691, size = 677, normalized size = 1.36 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (\frac{2 c^2 d \left (67 a e^2+79 c d^2\right )}{(d+e x) \left (a e^2+c d^2\right )}-\frac{4 c \left (4 a e^2+13 c d^2\right )}{(d+e x)^2}+\frac{18 c d \left (a e^2+c d^2\right )}{(d+e x)^3}-\frac{3 \left (a e^2+c d^2\right )^2}{(d+e x)^4}+7 c^2\right )}{e^5}-\frac{16 c^2 \left (-\sqrt{a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+8 i \sqrt{a} c d^2 e+29 a \sqrt{c} d e^2+32 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (29 a^2 e^2+a c \left (32 d^2+29 e^2 x^2\right )+32 c^2 d^2 x^2\right )+\sqrt{c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3+32 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2-32 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{e^7 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (a e^2+c d^2\right )}\right )}{21 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.286, size = 5303, normalized size = 10.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{\frac{9}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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