Optimal. Leaf size=94 \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0955831, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3/(c + d*x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 21.9889, size = 87, normalized size = 0.93 \[ \frac{2 b^{3} \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{2 b^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{d^{4}} + \frac{6 b \sqrt{c + d x} \left (a d - b c\right )^{2}}{d^{4}} - \frac{2 \left (a d - b c\right )^{3}}{d^{4} \sqrt{c + d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3/(d*x+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0852668, size = 99, normalized size = 1.05 \[ \frac{2 \left (-5 a^3 d^3+15 a^2 b d^2 (2 c+d x)+5 a b^2 d \left (-8 c^2-4 c d x+d^2 x^2\right )+b^3 \left (16 c^3+8 c^2 d x-2 c d^2 x^2+d^3 x^3\right )\right )}{5 d^4 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3/(c + d*x)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 116, normalized size = 1.2 \[ -{\frac{-2\,{b}^{3}{x}^{3}{d}^{3}-10\,a{b}^{2}{d}^{3}{x}^{2}+4\,{b}^{3}c{d}^{2}{x}^{2}-30\,{a}^{2}b{d}^{3}x+40\,a{b}^{2}c{d}^{2}x-16\,{b}^{3}{c}^{2}dx+10\,{a}^{3}{d}^{3}-60\,{a}^{2}bc{d}^{2}+80\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{5\,{d}^{4}}{\frac{1}{\sqrt{dx+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3/(d*x+c)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34382, size = 169, normalized size = 1.8 \[ \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{5}{2}} b^{3} - 5 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{d x + c}}{d^{3}} + \frac{5 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{3}}\right )}}{5 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/(d*x + c)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.208917, size = 154, normalized size = 1.64 \[ \frac{2 \,{\left (b^{3} d^{3} x^{3} + 16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} -{\left (2 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} +{\left (8 \, b^{3} c^{2} d - 20 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3}\right )} x\right )}}{5 \, \sqrt{d x + c} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/(d*x + c)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{3}}{\left (c + d x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3/(d*x+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.217979, size = 205, normalized size = 2.18 \[ \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{4}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{5}{2}} b^{3} d^{16} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c d^{16} + 15 \, \sqrt{d x + c} b^{3} c^{2} d^{16} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} d^{17} - 30 \, \sqrt{d x + c} a b^{2} c d^{17} + 15 \, \sqrt{d x + c} a^{2} b d^{18}\right )}}{5 \, d^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/(d*x + c)^(3/2),x, algorithm="giac")
[Out]