Optimal. Leaf size=201 \[ \frac{\sqrt{e} (-5 a B e+2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{7/2}}+\frac{e \sqrt{a+b x} \sqrt{d+e x} (-5 a B e+2 A b e+3 b B d)}{b^3 (b d-a e)}-\frac{2 (d+e x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.395576, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{e} (-5 a B e+2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{7/2}}+\frac{e \sqrt{a+b x} \sqrt{d+e x} (-5 a B e+2 A b e+3 b B d)}{b^3 (b d-a e)}-\frac{2 (d+e x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{3 b^2 \sqrt{a+b x} (b d-a e)}-\frac{2 (d+e x)^{5/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(3/2))/(a + b*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 37.7, size = 194, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a e - b d\right )} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (2 A b e - 5 B a e + 3 B b d\right )}{3 b^{2} \sqrt{a + b x} \left (a e - b d\right )} - \frac{e \sqrt{a + b x} \sqrt{d + e x} \left (2 A b e - 5 B a e + 3 B b d\right )}{b^{3} \left (a e - b d\right )} + \frac{\sqrt{e} \left (2 A b e - 5 B a e + 3 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(3/2)/(b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.296008, size = 149, normalized size = 0.74 \[ \frac{\sqrt{e} (-5 a B e+2 A b e+3 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{2 b^{7/2}}-\frac{\sqrt{d+e x} \left (B \left (-15 a^2 e+4 a b (d-5 e x)-3 b^2 x (e x-2 d)\right )+2 A b (3 a e+b (d+4 e x))\right )}{3 b^3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(3/2))/(a + b*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.033, size = 698, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(3/2)/(b*x+a)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(3/2)/(b*x + a)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.833445, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (3 \, B a^{2} b d +{\left (3 \, B b^{3} d -{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{2} -{\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} e + 2 \,{\left (3 \, B a b^{2} d -{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} e\right )} x\right )} \sqrt{\frac{e}{b}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} e x + b^{2} d + a b e\right )} \sqrt{b x + a} \sqrt{e x + d} \sqrt{\frac{e}{b}} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \,{\left (3 \, B b^{2} e x^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} e - 2 \,{\left (3 \, B b^{2} d - 2 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{12 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, \frac{3 \,{\left (3 \, B a^{2} b d +{\left (3 \, B b^{3} d -{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} e\right )} x^{2} -{\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} e + 2 \,{\left (3 \, B a b^{2} d -{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} e\right )} x\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, b e x + b d + a e}{2 \, \sqrt{b x + a} \sqrt{e x + d} b \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (3 \, B b^{2} e x^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d + 3 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} e - 2 \,{\left (3 \, B b^{2} d - 2 \,{\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{6 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(3/2)/(b*x + a)^(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((B*x+A)*(e*x+d)**(3/2)/(b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.633909, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(3/2)/(b*x + a)^(5/2),x, algorithm="giac")
[Out]