Optimal. Leaf size=666 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]
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Rubi [A] time = 3.01197, 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 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^3*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+a)**(1/2)/(g*x+f)**(1/2),x)
[Out]
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Mathematica [C] time = 13.5582, size = 5379, normalized size = 8.08 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^3*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]
[Out]
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Maple [B] time = 0.063, size = 5079, normalized size = 7.6 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{3}}{\sqrt{g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{2} + a}}{\sqrt{g x + f}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{2}} \left (d + e x\right )^{3}}{\sqrt{f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+a)**(1/2)/(g*x+f)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f),x, algorithm="giac")
[Out]