Optimal. Leaf size=491 \[ -\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{32 c^2 d \sqrt{a+c x^2} \left (2 a e^2+c d^2\right )}{35 e^3 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}+\frac{32 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a e^2+7 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.475794, antiderivative size = 491, 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, 835, 844, 719, 424, 419} \[ \frac{32 c^2 d \sqrt{a+c x^2} \left (2 a e^2+c d^2\right )}{35 e^3 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{32 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{35 e^4 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 c \sqrt{a+c x^2} \left (e x \left (5 a e^2+7 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{35 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 733
Rule 811
Rule 835
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac{(6 c) \int \frac{x \sqrt{a+c x^2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac{4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac{(4 c) \int \frac{3 a c d e-c \left (4 c d^2+5 a e^2\right ) x}{(d+e x)^{3/2} \sqrt{a+c x^2}} \, dx}{35 e^3 \left (c d^2+a e^2\right )}\\ &=\frac{32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}-\frac{4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac{(8 c) \int \frac{\frac{1}{2} a c e \left (c d^2+5 a e^2\right )-2 c^2 d \left (c d^2+2 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{35 e^3 \left (c d^2+a e^2\right )^2}\\ &=\frac{32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}-\frac{4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac{\left (16 c^3 d \left (c d^2+2 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{35 e^4 \left (c d^2+a e^2\right )^2}+\frac{\left (4 c^2 \left (4 c d^2+5 a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{35 e^4 \left (c d^2+a e^2\right )}\\ &=\frac{32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}-\frac{4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}-\frac{\left (32 a c^{5/2} d \left (c d^2+2 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 )}{35 \sqrt{-a} e^4 \left (c d^2+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (8 a c^{3/2} \left (4 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 )}{35 \sqrt{-a} e^4 \left (c d^2+a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{32 c^2 d \left (c d^2+2 a e^2\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right )^2 \sqrt{d+e x}}-\frac{4 c \left (2 d \left (2 c d^2+a e^2\right )+e \left (7 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{35 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}}+\frac{32 \sqrt{-a} c^{5/2} d \left (c d^2+2 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 )}{35 e^4 \left (c d^2+a e^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{8 \sqrt{-a} c^{3/2} \left (4 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 )}{35 e^4 \left (c d^2+a e^2\right ) \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 3.81197, size = 659, normalized size = 1.34 \[ \frac{2 \left (-e^2 \left (a+c x^2\right ) \left (-16 c^2 d (d+e x)^3 \left (2 a e^2+c d^2\right )-16 c d (d+e x) \left (a e^2+c d^2\right )^2+c (d+e x)^2 \left (15 a e^2+19 c d^2\right ) \left (a e^2+c d^2\right )+5 \left (a e^2+c d^2\right )^3\right )-\frac{4 c^2 (d+e x)^3 \left (-\sqrt{a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (2 a^2 e^2+a c \left (d^2+2 e^2 x^2\right )+c^2 d^2 x^2\right )+4 \sqrt{c} d (d+e x)^{3/2} \left (2 a^{3/2} e^3+\sqrt{a} c d^2 e-2 i a \sqrt{c} d e^2-i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{35 e^5 \sqrt{a+c x^2} (d+e x)^{7/2} \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.306, size = 5277, normalized size = 10.8 \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{3}{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 x^{2} + a\right )}^{\frac{3}{2}} \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{3}{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{3}{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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