Optimal. Leaf size=1257 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 4.33319, antiderivative size = 1257, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {940, 6742, 719, 419, 844, 424, 933, 168, 538, 537} \[ \frac{3 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{c x^2+a} e^2}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}-\frac{\sqrt{f+g x} \sqrt{c x^2+a} e^2}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 \sqrt{-a} \sqrt{c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{3 \sqrt{-a} \sqrt{c} f \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 ) e}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{3 \sqrt{-a} \sqrt{c} d g \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{\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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}}-\frac{3 \left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{c (e f-3 d g) \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 )}{\left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{c x^2+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 940
Rule 6742
Rule 719
Rule 419
Rule 844
Rule 424
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}} \, dx &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{\int \frac{3 a e^2 g-4 c d (e f-d g)+2 c e (e f-2 d g) x+c e^2 g x^2}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{\int \left (\frac{c g}{\sqrt{f+g x} \sqrt{a+c x^2}}+\frac{3 \left (a e^2 g-c d (2 e f-3 d g)\right )}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{2 c (e f-3 d g)}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}}\right ) \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}-\frac{(c g) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)}-\frac{(c (e f-3 d g)) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac{\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac{1}{(d+e x)^2 \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{4 \left (c d^2+a e^2\right ) (e f-d g)}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac{\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac{a e^2 g-2 c d (e f-d g)-2 c d e g x-c e^2 g x^2}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}-\frac{\left (c (e f-3 d g) \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}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{a+c x^2}}-\frac{\left (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 )}{2 \sqrt{-a} \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac{\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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \left (-\frac{c d g}{\sqrt{f+g x} \sqrt{a+c x^2}}-\frac{c e g x}{\sqrt{f+g x} \sqrt{a+c x^2}}+\frac{a e^2 g-c d (2 e f-3 d g)}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}}\right ) \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac{\left (c (e f-3 d 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{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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt{a+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac{\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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (3 c d g \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}-\frac{\left (3 c e g \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac{x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac{\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )^2\right ) \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac{\left (c (e f-3 d g) \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 )}{\left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac{\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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{c (e f-3 d g) \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 ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (3 c e \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac{\left (3 c e f \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2}+\frac{\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )^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}{8 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{a+c x^2}}-\frac{\left (3 a \sqrt{c} d g \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \sqrt{-a} \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac{\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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{3 \sqrt{-a} \sqrt{c} d g \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{c (e f-3 d g) \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 ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{a+c x^2}}-\frac{\left (3 a \sqrt{c} e \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \sqrt{-a} \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (3 a \sqrt{c} e f \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \sqrt{-a} \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac{3 \sqrt{-a} \sqrt{c} e \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{\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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{3 \sqrt{-a} \sqrt{c} e f \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{3 \sqrt{-a} \sqrt{c} d g \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{c (e f-3 d g) \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 ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{\left (3 \left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{e^2 \sqrt{f+g x} \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (e f-d g) (d+e x)^2}+\frac{3 e^2 \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 (d+e x)}+\frac{3 \sqrt{-a} \sqrt{c} e \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{\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 )}{2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{3 \sqrt{-a} \sqrt{c} e f \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{3 \sqrt{-a} \sqrt{c} d g \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}+\frac{c (e f-3 d g) \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 ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt{f+g x} \sqrt{a+c x^2}}-\frac{3 \left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g)^2 \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 12.3437, size = 2990, normalized size = 2.38 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.332, size = 20365, normalized size = 16.2 \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 )}^{3} \sqrt{g x + f}}\,{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{3} \sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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