Optimal. Leaf size=597 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{3 b^2 \sqrt{a+b x} (b e-a f)^2}+\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} (b e-a f)}+\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c \left (-2 A f^2+3 B e f+3 C e^2\right )+A d e f\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]
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Rubi [A] time = 3.82747, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{3 b^2 \sqrt{a+b x} (b e-a f)^2}+\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} (b e-a f)}+\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c f (3 B e-2 A f)+A d e f+3 c C e^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*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((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(5/2)/(f*x+e)**(1/2),x)
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Mathematica [C] time = 16.8165, size = 5074, normalized size = 8.5 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x]),x]
[Out]
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Maple [B] time = 0.115, size = 13614, normalized size = 22.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{f x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a} \sqrt{f x + e}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(5/2)/(f*x+e)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{f x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)),x, algorithm="giac")
[Out]