Optimal. Leaf size=163 \[ \frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 c (b-c)^3} \]
[Out]
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Rubi [A] time = 0.437136, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{2 (b+3 c) (a+b x)^{5/2}}{5 b^2 (b-c)^3}-\frac{2 a (b+3 c) (a+b x)^{3/2}}{3 b^2 (b-c)^3}-\frac{2 (3 b+c) (a+c x)^{5/2}}{5 c^2 (b-c)^3}+\frac{2 a (3 b+c) (a+c x)^{3/2}}{3 c^2 (b-c)^3}+\frac{8 a (a+b x)^{3/2}}{3 b (b-c)^3}-\frac{8 a (a+c x)^{3/2}}{3 c (b-c)^3} \]
Antiderivative was successfully verified.
[In] Int[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]
[Out]
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Rubi in Sympy [A] time = 35.1659, size = 144, normalized size = 0.88 \[ - \frac{8 a \left (a + c x\right )^{\frac{3}{2}}}{3 c \left (b - c\right )^{3}} + \frac{2 a \left (a + c x\right )^{\frac{3}{2}} \left (3 b + c\right )}{3 c^{2} \left (b - c\right )^{3}} + \frac{8 a \left (a + b x\right )^{\frac{3}{2}}}{3 b \left (b - c\right )^{3}} - \frac{2 a \left (a + b x\right )^{\frac{3}{2}} \left (b + 3 c\right )}{3 b^{2} \left (b - c\right )^{3}} - \frac{2 \left (a + c x\right )^{\frac{5}{2}} \left (3 b + c\right )}{5 c^{2} \left (b - c\right )^{3}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (b + 3 c\right )}{5 b^{2} \left (b - c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)
[Out]
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Mathematica [A] time = 0.340223, size = 80, normalized size = 0.49 \[ \frac{2 \left (c^2 (a+b x)^{3/2} (a (6 b-2 c)+b x (b+3 c))-b^2 (a+c x)^{3/2} (c x (3 b+c)-2 a (b-3 c))\right )}{5 b^2 c^2 (b-c)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]
[Out]
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Maple [A] time = 0.005, size = 172, normalized size = 1.1 \[ 2\,{\frac{1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a}{ \left ( b-c \right ) ^{3}b}}+{\frac{8\,a}{3\, \left ( b-c \right ) ^{3}b} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{8\,a}{3\, \left ( b-c \right ) ^{3}c} \left ( cx+a \right ) ^{{\frac{3}{2}}}}+6\,{\frac{c \left ( 1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a \right ) }{ \left ( b-c \right ) ^{3}{b}^{2}}}-6\,{\frac{b \left ( 1/5\, \left ( cx+a \right ) ^{5/2}-1/3\, \left ( cx+a \right ) ^{3/2}a \right ) }{ \left ( b-c \right ) ^{3}{c}^{2}}}-2\,{\frac{1/5\, \left ( cx+a \right ) ^{5/2}-1/3\, \left ( cx+a \right ) ^{3/2}a}{ \left ( b-c \right ) ^{3}c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273866, size = 225, normalized size = 1.38 \[ \frac{2 \,{\left ({\left (6 \, a^{2} b c^{2} - 2 \, a^{2} c^{3} +{\left (b^{3} c^{2} + 3 \, b^{2} c^{3}\right )} x^{2} +{\left (7 \, a b^{2} c^{2} + a b c^{3}\right )} x\right )} \sqrt{b x + a} +{\left (2 \, a^{2} b^{3} - 6 \, a^{2} b^{2} c -{\left (3 \, b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} -{\left (a b^{3} c + 7 \, a b^{2} c^{2}\right )} x\right )} \sqrt{c x + a}\right )}}{5 \,{\left (b^{5} c^{2} - 3 \, b^{4} c^{3} + 3 \, b^{3} c^{4} - b^{2} c^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(sqrt(b*x + a) + sqrt(c*x + a))^3,x, algorithm="giac")
[Out]