Optimal. Leaf size=319 \[ \frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} (a B e+A c d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]
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Rubi [A] time = 0.228865, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {821, 844, 719, 424, 419} \[ \frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} (a B e+A c d) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]
Antiderivative was successfully verified.
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Rule 821
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a B-A c x) \sqrt{d+e x}}{a c \sqrt{a+c x^2}}+\frac{\int \frac{\frac{a B e}{2}-\frac{1}{2} A c e x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a B-A c x) \sqrt{d+e x}}{a c \sqrt{a+c x^2}}-\frac{A \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{2 a}+\frac{(A c d+a B e) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{2 a c}\\ &=-\frac{(a B-A c x) \sqrt{d+e x}}{a c \sqrt{a+c x^2}}-\frac{\left (A \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 )}{\sqrt{-a} \sqrt{c} \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left ((A c d+a B e) \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 )}{\sqrt{-a} c^{3/2} \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=-\frac{(a B-A c x) \sqrt{d+e x}}{a c \sqrt{a+c x^2}}-\frac{A \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 )}{\sqrt{-a} \sqrt{c} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}+\frac{(A c d+a B e) \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 )}{\sqrt{-a} c^{3/2} \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 4.16078, size = 431, normalized size = 1.35 \[ \frac{\sqrt{d+e x} \left (\frac{\sqrt{a} \sqrt{d+e x} \left (A \sqrt{c}+i \sqrt{a} B\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{A e \left (a+c x^2\right )}{d+e x}-\frac{i A c \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e}-a B+A c x\right )}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 979, normalized size = 3.1 \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 (B x + A\right )} \sqrt{e x + d}}{{\left (c x^{2} + a\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 (\frac{\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x + d}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, 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 + B x\right ) \sqrt{d + e x}}{\left (a + c x^{2}\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 \frac{{\left (B x + A\right )} \sqrt{e x + d}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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