Optimal. Leaf size=448 \[ -\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{315 e^3}+\frac{2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{4 d \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{21 e} \]
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Rubi [A] time = 0.467111, antiderivative size = 448, 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 = {735, 833, 815, 844, 719, 424, 419} \[ \frac{8 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{4 \sqrt{a+c x^2} \sqrt{d+e x} \left (4 d \left (3 a e^2+c d^2\right )-3 e x \left (c d^2-7 a e^2\right )\right )}{315 e^3}-\frac{32 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{315 \sqrt{c} e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \left (a+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{4 d \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{21 e} \]
Antiderivative was successfully verified.
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Rule 735
Rule 833
Rule 815
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a+c x^2\right )^{3/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{2 \int (a e-c d x) \sqrt{d+e x} \sqrt{a+c x^2} \, dx}{3 e}\\ &=-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{4 \int \frac{\left (4 a c d e-\frac{1}{2} c \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{16 \int \frac{\frac{1}{4} a c^2 d e \left (c d^2+33 a e^2\right )-\frac{1}{4} c^2 \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{315 c^2 e^3}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{\left (16 d \left (c d^2+a e^2\right ) \left (c d^2+3 a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{315 e^4}-\frac{\left (4 \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{315 e^4}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}-\frac{\left (8 a \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{315 \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 (32 a d \left (c d^2+a e^2\right ) \left (c d^2+3 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 )}{315 \sqrt{-a} \sqrt{c} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{4 \sqrt{d+e x} \left (4 d \left (c d^2+3 a e^2\right )-3 e \left (c d^2-7 a e^2\right ) x\right ) \sqrt{a+c x^2}}{315 e^3}-\frac{4 d \sqrt{d+e x} \left (a+c x^2\right )^{3/2}}{21 e}+\frac{2 (d+e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 e}+\frac{8 \sqrt{-a} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\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 )}{315 \sqrt{c} e^4 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{32 \sqrt{-a} d \left (c d^2+a e^2\right ) \left (c d^2+3 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 )}{315 \sqrt{c} e^4 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 4.16675, size = 646, normalized size = 1.44 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (a e^2 (29 d+77 e x)+c \left (-6 d^2 e x+8 d^3+5 d e^2 x^2+35 e^3 x^3\right )\right )}{e^3}+\frac{8 \left (\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (33 i a^{3/2} \sqrt{c} d e^3-21 a^2 e^4+i \sqrt{a} c^{3/2} d^3 e+15 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )+\sqrt{c} (d+e x)^{3/2} \left (-15 a^{3/2} c d^2 e^3-21 i a^2 \sqrt{c} d e^4+21 a^{5/2} e^5+15 i a c^{3/2} d^3 e^2-4 \sqrt{a} c^2 d^4 e+4 i c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c e^5 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{315 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.25, size = 1731, normalized size = 3.9 \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}} \sqrt{e x + d}\,{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 x^{2} + a\right )}^{\frac{3}{2}} \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}} \sqrt{d + e x}\, 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}} \sqrt{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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