Optimal. Leaf size=666 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (168 d^2 e f g^2-35 d^3 g^3-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]
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Rubi [A] time = 1.57141, antiderivative size = 666, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {921, 1654, 844, 719, 424, 419} \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (168 d^2 e f g^2-35 d^3 g^3-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]
Antiderivative was successfully verified.
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Rule 921
Rule 1654
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}-\frac{\int \frac{(d+e x)^2 \left (2 a (3 e f-4 d g)+2 (c d f-a e g) x+2 c (4 e f-3 d g) x^2\right )}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{9 g}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{2 \int \frac{-a c g^2 \left (20 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+28 d^3 g^3\right )-c g \left (a e g^2 \left (40 e^2 f^2-72 d e f g+63 d^2 g^2\right )+c \left (8 e^3 f^4-6 d e^2 f^3 g-7 d^3 f g^3\right )\right ) x+c g^2 \left (a e^2 g^2 (e f-27 d g)-c \left (44 e^3 f^3-33 d e^2 f^2 g-42 d^2 e f g^2+21 d^3 g^3\right )\right ) x^2-c e g^3 \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) x^3}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{63 c g^5}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{4 \int \frac{\frac{1}{2} a c g^5 \left (21 a e^3 f g^2+c \left (92 e^3 f^3-258 d e^2 f^2 g+231 d^2 e f g^2-140 d^3 g^3\right )\right )+\frac{1}{2} c g^4 \left (21 a^2 e^3 g^4+3 a c e g^2 \left (2 e^2 f^2+9 d e f g-63 d^2 g^2\right )+c^2 f \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )\right ) x+\frac{3}{2} c^2 g^5 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) x^2}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{315 c^2 g^8}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{8 \int \frac{\frac{3}{4} a c^2 g^7 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )+\frac{3}{4} c^2 g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{945 c^3 g^{10}}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{\left (2 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{315 c g^5}-\frac{\left (8 \left (\frac{3}{4} a c^2 g^8 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-\frac{3}{4} c^2 f g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{945 c^3 g^{11}}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{\left (4 a \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\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 )}{315 \sqrt{-a} c^{3/2} g^5 \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{a+c x^2}}-\frac{\left (16 a \left (\frac{3}{4} a c^2 g^8 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-\frac{3}{4} c^2 f g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\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 )}{945 \sqrt{-a} c^{7/2} g^{11} \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}+\frac{4 \sqrt{-a} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\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 )}{315 c^{3/2} g^5 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{a+c x^2}}+\frac{4 \sqrt{-a} \left (c f^2+a g^2\right ) \left (64 c e^3 f^3-216 c d e^2 f^2 g+252 c d^2 e f g^2-18 a e^3 f g^2-105 c d^3 g^3+45 a d e^2 g^3\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 )}{315 c^{3/2} g^5 \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 8.1031, size = 864, normalized size = 1.3 \[ \frac{2 \sqrt{f+g x} \left (-c \left (c x^2+a\right ) \left (c \left (\left (64 f^3-48 g x f^2+40 g^2 x^2 f-35 g^3 x^3\right ) e^3-27 d g \left (8 f^2-6 g x f+5 g^2 x^2\right ) e^2+63 d^2 g^2 (4 f-3 g x) e-105 d^3 g^3\right )-2 a e^2 g^2 (-11 e f+45 d g+7 e g x)\right ) g^2-\frac{2 \left (\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2+c^2 f \left (-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right )\right ) \left (c x^2+a\right ) g^2+\sqrt{a} \sqrt{c} \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (21 i a^{3/2} e^3 g^3-9 a \sqrt{c} e^2 (2 e f-5 d g) g^2-3 i \sqrt{a} c e \left (16 e^2 f^2-54 d e g f+63 d^2 g^2\right ) g+c^{3/2} \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right ) g-\sqrt{c} \left (i \sqrt{c} f-\sqrt{a} g\right ) \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2+c^2 f \left (-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right )\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)}\right )}{315 c^2 g^6 \sqrt{c x^2+a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.287, size = 5079, normalized size = 7.6 \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{\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{2} + a}}{\sqrt{g x + f}}, 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{\sqrt{a + c x^{2}} \left (d + e x\right )^{3}}{\sqrt{f + g 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 \frac{\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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