Optimal. Leaf size=818 \[ -\frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right ) e^2}{\left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right ) e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}+\frac{2 g^2 \sqrt{c x^2+a} e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{8 \sqrt{-a} c^{3/2} f g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{8 c f g^2 \sqrt{c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 g^2 \sqrt{c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}} \]
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Rubi [A] time = 0.996386, antiderivative size = 818, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {958, 745, 835, 844, 719, 424, 419, 21, 933, 168, 538, 537} \[ -\frac{2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right ) e^2}{\left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right ) e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}+\frac{2 g^2 \sqrt{c x^2+a} e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{8 \sqrt{-a} c^{3/2} f g \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{8 c f g^2 \sqrt{c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 g^2 \sqrt{c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 958
Rule 745
Rule 835
Rule 844
Rule 719
Rule 424
Rule 419
Rule 21
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) (f+g x)^{5/2} \sqrt{a+c x^2}} \, dx &=\int \left (-\frac{g}{(e f-d g) (f+g x)^{5/2} \sqrt{a+c x^2}}-\frac{e g}{(e f-d g)^2 (f+g x)^{3/2} \sqrt{a+c x^2}}+\frac{e^2}{(e f-d g)^2 (d+e x) \sqrt{f+g x} \sqrt{a+c x^2}}\right ) \, dx\\ &=\frac{e^2 \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{(e f-d g)^2}-\frac{(e g) \int \frac{1}{(f+g x)^{3/2} \sqrt{a+c x^2}} \, dx}{(e f-d g)^2}-\frac{g \int \frac{1}{(f+g x)^{5/2} \sqrt{a+c x^2}} \, dx}{e f-d g}\\ &=\frac{2 g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac{2 e g^2 \sqrt{a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{(2 c e g) \int \frac{-\frac{f}{2}-\frac{g x}{2}}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}+\frac{(2 c g) \int \frac{-\frac{3 f}{2}+\frac{g x}{2}}{(f+g x)^{3/2} \sqrt{a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}+\frac{\left (e^2 \sqrt{1+\frac{c x^2}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}} \sqrt{1+\frac{\sqrt{c} x}{\sqrt{-a}}} (d+e x) \sqrt{f+g x}} \, dx}{(e f-d g)^2 \sqrt{a+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac{8 c f g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{(4 c g) \int \frac{\frac{1}{4} \left (3 c f^2-a g^2\right )+c f g x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}-\frac{(c e g) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}-\frac{\left (2 e^2 \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e-e x^2\right ) \sqrt{f+\frac{\sqrt{-a} g}{\sqrt{c}}-\frac{\sqrt{-a} g x^2}{\sqrt{c}}}} \, dx,x,\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}\right )}{(e f-d g)^2 \sqrt{a+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac{8 c f g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{\left (4 c^2 f g\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}+\frac{(c g) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}-\frac{\left (2 e^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e-e x^2\right ) \sqrt{1-\frac{\sqrt{-a} g x^2}{\sqrt{c} \left (f+\frac{\sqrt{-a} g}{\sqrt{c}}\right )}}} \, dx,x,\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}\right )}{(e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (2 a \sqrt{c} e g \sqrt{f+g x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} g x^2}{\sqrt{-a} \left (c f-\frac{a \sqrt{c} g}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{\sqrt{-a} (e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac{8 c f g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{2 \sqrt{-a} \sqrt{c} e g \sqrt{f+g 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 g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}-\frac{2 e^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{1+\frac{c x^2}{a}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (8 a c^{3/2} f g \sqrt{f+g x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} g x^2}{\sqrt{-a} \left (c f-\frac{a \sqrt{c} g}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (2 a \sqrt{c} g \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\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} g x^2}{\sqrt{-a} \left (c f-\frac{a \sqrt{c} g}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} (e f-d g) \left (c f^2+a g^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=\frac{2 g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac{8 c f g^2 \sqrt{a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{f+g x}}+\frac{2 e g^2 \sqrt{a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{8 \sqrt{-a} c^{3/2} f g \sqrt{f+g 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 g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{2 \sqrt{-a} \sqrt{c} e g \sqrt{f+g 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 g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}-\frac{2 \sqrt{-a} \sqrt{c} g \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \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 g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{2 e^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{1+\frac{c x^2}{a}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 9.23841, size = 1917, normalized size = 2.34 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.334, size = 9409, normalized size = 11.5 \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{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\sqrt{a + c x^{2}} \left (d + e x\right ) \left (f + g x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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