Optimal. Leaf size=89 \[ \frac{2 \sqrt{d+e x} (a B e-2 A b e+b B d)}{e \sqrt{a+b x} (b d-a e)^2}-\frac{2 (B d-A e)}{e \sqrt{a+b x} \sqrt{d+e x} (b d-a e)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.178216, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{d+e x} (a B e-2 A b e+b B d)}{e \sqrt{a+b x} (b d-a e)^2}-\frac{2 (B d-A e)}{e \sqrt{a+b x} \sqrt{d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.0069, size = 78, normalized size = 0.88 \[ \frac{4 \sqrt{a + b x} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{b \sqrt{d + e x} \left (a e - b d\right )^{2}} + \frac{2 \left (A b - B a\right )}{b \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.130139, size = 61, normalized size = 0.69 \[ \frac{2 B (2 a d+a e x+b d x)-2 A (a e+b (d+2 e x))}{\sqrt{a+b x} \sqrt{d+e x} (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((a + b*x)^(3/2)*(d + e*x)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 72, normalized size = 0.8 \[ -2\,{\frac{2\,Abex-Baex-Bbdx+Aae+Abd-2\,Bad}{\sqrt{bx+a}\sqrt{ex+d} \left ({a}^{2}{e}^{2}-2\,bead+{b}^{2}{d}^{2} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b*x+a)^(3/2)/(e*x+d)^(3/2),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)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.481613, size = 200, normalized size = 2.25 \[ -\frac{2 \,{\left (A a e -{\left (2 \, B a - A b\right )} d -{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} +{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b*x+a)**(3/2)/(e*x+d)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.258091, size = 227, normalized size = 2.55 \[ \frac{2 \,{\left (B b^{2} d - A b^{2} e\right )} \sqrt{b x + a}}{{\left (b^{2} d^{2}{\left | b \right |} - 2 \, a b d{\left | b \right |} e + a^{2}{\left | b \right |} e^{2}\right )} \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} + \frac{4 \,{\left (B a b^{\frac{3}{2}} e^{\frac{1}{2}} - A b^{\frac{5}{2}} e^{\frac{1}{2}}\right )}}{{\left (b^{2} d - a b e -{\left (\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{\left (b d{\left | b \right |} - a{\left | b \right |} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="giac")
[Out]