Optimal. Leaf size=359 \[ \frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{3/2}}{5 c}+\frac{16 d e \sqrt{a+c x^2} \sqrt{d+e x}}{15 c} \]
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Rubi [A] time = 0.32498, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {743, 833, 844, 719, 424, 419} \[ \frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{3/2}}{5 c}+\frac{16 d e \sqrt{a+c x^2} \sqrt{d+e x}}{15 c} \]
Antiderivative was successfully verified.
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Rule 743
Rule 833
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{\sqrt{a+c x^2}} \, dx &=\frac{2 e (d+e x)^{3/2} \sqrt{a+c x^2}}{5 c}+\frac{2 \int \frac{\sqrt{d+e x} \left (\frac{1}{2} \left (5 c d^2-3 a e^2\right )+4 c d e x\right )}{\sqrt{a+c x^2}} \, dx}{5 c}\\ &=\frac{16 d e \sqrt{d+e x} \sqrt{a+c x^2}}{15 c}+\frac{2 e (d+e x)^{3/2} \sqrt{a+c x^2}}{5 c}+\frac{4 \int \frac{\frac{1}{4} c d \left (15 c d^2-17 a e^2\right )+\frac{1}{4} c e \left (23 c d^2-9 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{15 c^2}\\ &=\frac{16 d e \sqrt{d+e x} \sqrt{a+c x^2}}{15 c}+\frac{2 e (d+e x)^{3/2} \sqrt{a+c x^2}}{5 c}+\frac{\left (23 c d^2-9 a e^2\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{15 c}-\frac{\left (8 d \left (c d^2+a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{15 c}\\ &=\frac{16 d e \sqrt{d+e x} \sqrt{a+c x^2}}{15 c}+\frac{2 e (d+e x)^{3/2} \sqrt{a+c x^2}}{5 c}+\frac{\left (2 a \left (23 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 )}{15 \sqrt{-a} c^{3/2} \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}-\frac{\left (16 a d \left (c d^2+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 )}{15 \sqrt{-a} c^{3/2} \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{16 d e \sqrt{d+e x} \sqrt{a+c x^2}}{15 c}+\frac{2 e (d+e x)^{3/2} \sqrt{a+c x^2}}{5 c}-\frac{2 \sqrt{-a} \left (23 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 )}{15 c^{3/2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}+\frac{16 \sqrt{-a} d \left (c d^2+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 )}{15 c^{3/2} \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 3.05718, size = 557, normalized size = 1.55 \[ \frac{\sqrt{d+e x} \left (\frac{2 e \left (a+c x^2\right ) (11 d+3 e x)}{c}+\frac{2 \left (\sqrt{c} (d+e x)^{3/2} \left (9 a^{3/2} e^3-23 \sqrt{a} c d^2 e-17 i a \sqrt{c} d e^2+15 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-9 a^2 e^2+a c \left (23 d^2-9 e^2 x^2\right )+23 c^2 d^2 x^2\right )+\sqrt{c} (d+e x)^{3/2} \left (-9 a^{3/2} e^3+23 \sqrt{a} c d^2 e+9 i a \sqrt{c} d e^2-23 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c^2 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{15 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.287, size = 1312, normalized size = 3.7 \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 \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + a}}\,{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 (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + a}}, 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 (d + e x\right )^{\frac{5}{2}}}{\sqrt{a + c x^{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 (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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