Optimal. Leaf size=497 \[ -\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}+\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]
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Rubi [A] time = 0.559344, antiderivative size = 497, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {743, 815, 844, 719, 424, 419} \[ \frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{1155 c e^3}-\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{231 c e}+\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{1155 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]
Antiderivative was successfully verified.
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Rule 743
Rule 815
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx &=\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{2 \int \frac{\left (\frac{1}{2} \left (11 c d^2-a e^2\right )+6 c d e x\right ) \left (a+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 c}\\ &=\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{8 \int \frac{\left (\frac{3}{4} a c e^2 \left (29 c d^2-3 a e^2\right )-\frac{3}{4} c^2 d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{32 \int \frac{\frac{3}{8} a c^2 e^2 \left (c^2 d^4+114 a c d^2 e^2-15 a^2 e^4\right )-\frac{3}{2} c^3 d e \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac{\left (16 d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{1155 e^4}+\frac{\left (4 \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{1155 c e^4}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac{\left (32 a d \left (c d^2-3 a e^2\right ) \left (c d^2+9 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 )}{1155 \sqrt{-a} \sqrt{c} e^4 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (8 a \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\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 )}{1155 \sqrt{-a} c^{3/2} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{4 \sqrt{d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt{a+c x^2}}{1155 c e^3}+\frac{2 \sqrt{d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac{2 e \sqrt{d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac{32 \sqrt{-a} d \left (c d^2-3 a e^2\right ) \left (c d^2+9 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 )}{1155 \sqrt{c} e^4 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{8 \sqrt{-a} \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\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 )}{1155 c^{3/2} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 3.98236, size = 695, normalized size = 1.4 \[ \frac{2 \sqrt{d+e x} \left (e^2 \left (a+c x^2\right ) \left (60 a^2 e^4+a c e^2 \left (47 d^2+326 d e x+195 e^2 x^2\right )+c^2 \left (5 d^2 e^2 x^2-6 d^3 e x+8 d^4+140 d e^3 x^3+105 e^4 x^4\right )\right )+\frac{4 \left (\sqrt{a} e (d+e x)^{3/2} \left (114 i a^{3/2} c d^2 e^3-108 a^2 \sqrt{c} d e^4-15 i a^{5/2} e^5+24 a c^{3/2} d^3 e^2+i \sqrt{a} c^2 d^4 e+4 c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )-4 d e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-27 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 \sqrt{c} d (d+e x)^{3/2} \left (-6 a^{3/2} c d^2 e^3-27 i a^2 \sqrt{c} d e^4+27 a^{5/2} e^5+6 i a c^{3/2} d^3 e^2-\sqrt{a} c^2 d^4 e+i c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{(d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{1155 c e^5 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.274, size = 1970, normalized size = 4. \begin{align*} \text{result 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{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{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 ({\left (c e x^{3} + c d x^{2} + a e x + a d\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}, 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 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{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{\left (c x^{2} + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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