Optimal. Leaf size=151 \[ \frac{2 \sqrt{a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt{c+d x} (b c-a d)^3}-\frac{2 \sqrt{a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.345701, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2 \sqrt{a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt{c+d x} (b c-a d)^3}-\frac{2 \sqrt{a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^2}-\frac{2 a^2}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 26.1966, size = 131, normalized size = 0.87 \[ - \frac{2 c^{2}}{3 d^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 c \left (3 a d - b c\right )}{3 d^{2} \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{2 \sqrt{c + d x} \left (3 a^{2} d^{2} + 6 a b c d - b^{2} c^{2}\right )}{3 d^{2} \sqrt{a + b x} \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.160664, size = 82, normalized size = 0.54 \[ \frac{-2 a^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )-4 a b c x (2 c+3 d x)+2 b^2 c^2 x^2}{3 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x)^(3/2)*(c + d*x)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 111, normalized size = 0.7 \[{\frac{6\,{a}^{2}{d}^{2}{x}^{2}+12\,abcd{x}^{2}-2\,{b}^{2}{c}^{2}{x}^{2}+24\,{a}^{2}cdx+8\,ab{c}^{2}x+16\,{a}^{2}{c}^{2}}{3\,{a}^{3}{d}^{3}-9\,{a}^{2}cb{d}^{2}+9\,a{b}^{2}{c}^{2}d-3\,{b}^{3}{c}^{3}}{\frac{1}{\sqrt{bx+a}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^(3/2)/(d*x+c)^(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(x^2/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.366875, size = 371, normalized size = 2.46 \[ -\frac{2 \,{\left (8 \, a^{2} c^{2} -{\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} + 4 \,{\left (a b c^{2} + 3 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x+a)**(3/2)/(d*x+c)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.289086, size = 423, normalized size = 2.8 \[ -\frac{4 \, \sqrt{b d} a^{2} b}{{\left (b^{2} c^{2}{\left | b \right |} - 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} - \frac{\sqrt{b x + a}{\left (\frac{{\left (b^{6} c^{4} d{\left | b \right |} - 8 \, a b^{5} c^{3} d^{2}{\left | b \right |} + 13 \, a^{2} b^{4} c^{2} d^{3}{\left | b \right |} - 6 \, a^{3} b^{3} c d^{4}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} - \frac{6 \,{\left (a b^{6} c^{4} d{\left | b \right |} - 3 \, a^{2} b^{5} c^{3} d^{2}{\left | b \right |} + 3 \, a^{3} b^{4} c^{2} d^{3}{\left | b \right |} - a^{4} b^{3} c d^{4}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^(3/2)*(d*x + c)^(5/2)),x, algorithm="giac")
[Out]