Optimal. Leaf size=346 \[ -\frac{2 \sqrt{b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (-2 A c e-3 b B e+8 B c d) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]
[Out]
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Rubi [A] time = 1.06537, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \sqrt{b x+c x^2} \left (d^2 (-A c e-3 b B e+4 B c d)+e x (B d (5 c d-4 b e)-A e (2 c d-b e))\right )}{3 d e^2 (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (-2 A c e-3 b B e+8 B c d) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^3 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (B d (8 c d-7 b e)-A e (2 c d-b e)) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 110.859, size = 326, normalized size = 0.94 \[ - \frac{2 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e \left (A e - 7 B d\right ) - 2 c d \left (A e - 4 B d\right )\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d e^{3} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- d} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (2 A c e + 3 B b e - 8 B c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 e^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{4 \sqrt{b x + c x^{2}} \left (\frac{d^{2} \left (A c e + 3 B b e - 4 B c d\right )}{2} - \frac{e x \left (A b e^{2} - 2 A c d e - 4 B b d e + 5 B c d^{2}\right )}{2}\right )}{3 d e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [C] time = 2.58296, size = 346, normalized size = 1. \[ \frac{2 \left (e x \sqrt{\frac{b}{c}} (b+c x) \left (A e \left (c d (d+2 e x)-b e^2 x\right )+B d (b e (3 d+4 e x)-c d (4 d+5 e x))\right )+(d+e x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (4 B d-A e) (c d-b e) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (A e (2 c d-b e)+B d (7 b e-8 c d)) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) (A e (b e-2 c d)+B d (8 c d-7 b e))\right )\right )}{3 d e^3 \sqrt{\frac{b}{c}} \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^(5/2),x]
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Maple [B] time = 0.048, size = 1959, normalized size = 5.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x}{\left (B x + A\right )}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]