Optimal. Leaf size=186 \[ -\frac{2 (B d-A e)}{3 e (a+b x)^{3/2} (d+e x)^{3/2} (b d-a e)}-\frac{16 e \sqrt{a+b x} (a B e-2 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^4}-\frac{8 (a B e-2 A b e+b B d)}{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3}+\frac{2 (a B e-2 A b e+b B d)}{3 e (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.351836, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 (B d-A e)}{3 e (a+b x)^{3/2} (d+e x)^{3/2} (b d-a e)}-\frac{16 e \sqrt{a+b x} (a B e-2 A b e+b B d)}{3 \sqrt{d+e x} (b d-a e)^4}-\frac{8 (a B e-2 A b e+b B d)}{3 \sqrt{a+b x} \sqrt{d+e x} (b d-a e)^3}+\frac{2 (a B e-2 A b e+b B d)}{3 e (a+b x)^{3/2} \sqrt{d+e x} (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 33.8214, size = 177, normalized size = 0.95 \[ \frac{16 e \sqrt{a + b x} \left (2 A b e - B a e - B b d\right )}{3 \sqrt{d + e x} \left (a e - b d\right )^{4}} - \frac{8 e \sqrt{a + b x} \left (2 A b e - B a e - B b d\right )}{3 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} - \frac{4 \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{b \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**(5/2)/(e*x+d)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.593051, size = 135, normalized size = 0.73 \[ \frac{2 \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{b (A b-a B) (b d-a e)}{(a+b x)^2}+\frac{e (b d-a e) (A e-B d)}{(d+e x)^2}-\frac{b (5 a B e-8 A b e+3 b B d)}{a+b x}+\frac{e (-3 a B e+8 A b e-5 b B d)}{d+e x}\right )}{3 (b d-a e)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x)^(5/2)*(d + e*x)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.013, size = 320, normalized size = 1.7 \[ -{\frac{-32\,A{b}^{3}{e}^{3}{x}^{3}+16\,Ba{b}^{2}{e}^{3}{x}^{3}+16\,B{b}^{3}d{e}^{2}{x}^{3}-48\,Aa{b}^{2}{e}^{3}{x}^{2}-48\,A{b}^{3}d{e}^{2}{x}^{2}+24\,B{a}^{2}b{e}^{3}{x}^{2}+48\,Ba{b}^{2}d{e}^{2}{x}^{2}+24\,B{b}^{3}{d}^{2}e{x}^{2}-12\,A{a}^{2}b{e}^{3}x-72\,Aa{b}^{2}d{e}^{2}x-12\,A{b}^{3}{d}^{2}ex+6\,B{a}^{3}{e}^{3}x+42\,B{a}^{2}bd{e}^{2}x+42\,Ba{b}^{2}{d}^{2}ex+6\,B{b}^{3}{d}^{3}x+2\,A{a}^{3}{e}^{3}-18\,A{a}^{2}bd{e}^{2}-18\,Aa{b}^{2}{d}^{2}e+2\,A{b}^{3}{d}^{3}+4\,B{a}^{3}d{e}^{2}+24\,B{a}^{2}b{d}^{2}e+4\,Ba{b}^{2}{d}^{3}}{3\,{e}^{4}{a}^{4}-12\,b{e}^{3}d{a}^{3}+18\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-12\,a{b}^{3}{d}^{3}e+3\,{b}^{4}{d}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^(5/2)/(e*x+d)^(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)/((b*x + a)^(5/2)*(e*x + d)^(5/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 1.11949, size = 763, normalized size = 4.1 \[ -\frac{2 \,{\left (A a^{3} e^{3} +{\left (2 \, B a b^{2} + A b^{3}\right )} d^{3} + 3 \,{\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e +{\left (2 \, B a^{3} - 9 \, A a^{2} b\right )} d e^{2} + 8 \,{\left (B b^{3} d e^{2} +{\left (B a b^{2} - 2 \, A b^{3}\right )} e^{3}\right )} x^{3} + 12 \,{\left (B b^{3} d^{2} e + 2 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} +{\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3}\right )} x^{2} + 3 \,{\left (B b^{3} d^{3} +{\left (7 \, B a b^{2} - 2 \, A b^{3}\right )} d^{2} e +{\left (7 \, B a^{2} b - 12 \, A a b^{2}\right )} d e^{2} +{\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} +{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} +{\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \,{\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(5/2)*(e*x + d)^(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)/(b*x+a)**(5/2)/(e*x+d)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.70033, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(5/2)*(e*x + d)^(5/2)),x, algorithm="giac")
[Out]