Optimal. Leaf size=344 \[ \frac{2 \sqrt{d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{3 b^4 \sqrt{b x+c x^2} (c d-b e)}-\frac{2 \sqrt{d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.22425, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{2 \sqrt{d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{3 b^4 \sqrt{b x+c x^2} (c d-b e)}-\frac{2 \sqrt{d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 158.682, size = 294, normalized size = 0.85 \[ \frac{16 \sqrt{c} \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \left (- b\right )^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 4 c d\right ) \left (3 b e - 4 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} \left (- b\right )^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{d + e x} \left (b d - x \left (b e - 2 c d\right )\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{4 \sqrt{d + e x} \left (\frac{b \left (5 b e - 8 c d\right )}{2} + 4 c x \left (b e - 2 c d\right )\right )}{3 b^{4} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.6735, size = 290, normalized size = 0.84 \[ \frac{2 \left (\frac{(d+e x) \left (b^3 (-(d+4 e x))+b^2 c x (6 d-13 e x)-8 b c^2 x^2 (e x-3 d)+16 c^3 d x^3\right )}{b+c x}+i c e x^{5/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (8 c d-5 b e) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+8 i c e x^{5/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-8 x (b+c x) (d+e x) (2 c d-b e)\right )}{3 b^4 x \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.045, size = 1099, normalized size = 3.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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((e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]