Optimal. Leaf size=261 \[ \frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac{8 c^2 (b x+c)^{3/2}}{3 b^2 (a-c)^3}+\frac{8 (a+b x)^{7/2}}{7 b^2 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac{16 a (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac{2 a (a+3 c) (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac{8 (b x+c)^{7/2}}{7 b^2 (a-c)^3}+\frac{16 c (b x+c)^{5/2}}{5 b^2 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{5/2}}{5 b^2 (a-c)^3}+\frac{2 c (3 a+c) (b x+c)^{3/2}}{3 b^2 (a-c)^3} \]
[Out]
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Rubi [A] time = 0.50518, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{8 a^2 (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac{8 c^2 (b x+c)^{3/2}}{3 b^2 (a-c)^3}+\frac{8 (a+b x)^{7/2}}{7 b^2 (a-c)^3}+\frac{2 (a+3 c) (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac{16 a (a+b x)^{5/2}}{5 b^2 (a-c)^3}-\frac{2 a (a+3 c) (a+b x)^{3/2}}{3 b^2 (a-c)^3}-\frac{8 (b x+c)^{7/2}}{7 b^2 (a-c)^3}+\frac{16 c (b x+c)^{5/2}}{5 b^2 (a-c)^3}-\frac{2 (3 a+c) (b x+c)^{5/2}}{5 b^2 (a-c)^3}+\frac{2 c (3 a+c) (b x+c)^{3/2}}{3 b^2 (a-c)^3} \]
Antiderivative was successfully verified.
[In] Int[x/(Sqrt[a + b*x] + Sqrt[c + b*x])^3,x]
[Out]
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Rubi in Sympy [A] time = 53.0561, size = 236, normalized size = 0.9 \[ \frac{8 a^{2} \left (a + b x\right )^{\frac{3}{2}}}{3 b^{2} \left (a - c\right )^{3}} - \frac{2 a \left (a + 3 c\right ) \left (a + b x\right )^{\frac{3}{2}}}{3 b^{2} \left (a - c\right )^{3}} - \frac{16 a \left (a + b x\right )^{\frac{5}{2}}}{5 b^{2} \left (a - c\right )^{3}} - \frac{8 c^{2} \left (b x + c\right )^{\frac{3}{2}}}{3 b^{2} \left (a - c\right )^{3}} + \frac{2 c \left (3 a + c\right ) \left (b x + c\right )^{\frac{3}{2}}}{3 b^{2} \left (a - c\right )^{3}} + \frac{16 c \left (b x + c\right )^{\frac{5}{2}}}{5 b^{2} \left (a - c\right )^{3}} + \frac{2 \left (a + 3 c\right ) \left (a + b x\right )^{\frac{5}{2}}}{5 b^{2} \left (a - c\right )^{3}} + \frac{8 \left (a + b x\right )^{\frac{7}{2}}}{7 b^{2} \left (a - c\right )^{3}} - \frac{2 \left (3 a + c\right ) \left (b x + c\right )^{\frac{5}{2}}}{5 b^{2} \left (a - c\right )^{3}} - \frac{8 \left (b x + c\right )^{\frac{7}{2}}}{7 b^{2} \left (a - c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/((b*x+a)**(1/2)+(b*x+c)**(1/2))**3,x)
[Out]
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Mathematica [A] time = 0.397595, size = 93, normalized size = 0.36 \[ \frac{2 \left ((a+b x)^{3/2} \left (6 a^2-a (9 b x+14 c)+b x (20 b x+21 c)\right )+(b x+c)^{3/2} \left (7 a (2 c-3 b x)-20 b^2 x^2+9 b c x-6 c^2\right )\right )}{35 b^2 (a-c)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x/(Sqrt[a + b*x] + Sqrt[c + b*x])^3,x]
[Out]
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Maple [A] time = 0.004, size = 222, normalized size = 0.9 \[ 2\,{\frac{a \left ( 1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a \right ) }{ \left ( a-c \right ) ^{3}{b}^{2}}}+6\,{\frac{c \left ( 1/5\, \left ( bx+a \right ) ^{5/2}-1/3\, \left ( bx+a \right ) ^{3/2}a \right ) }{ \left ( a-c \right ) ^{3}{b}^{2}}}-6\,{\frac{a \left ( 1/5\, \left ( bx+c \right ) ^{5/2}-1/3\, \left ( bx+c \right ) ^{3/2}c \right ) }{ \left ( a-c \right ) ^{3}{b}^{2}}}-2\,{\frac{c \left ( 1/5\, \left ( bx+c \right ) ^{5/2}-1/3\, \left ( bx+c \right ) ^{3/2}c \right ) }{ \left ( a-c \right ) ^{3}{b}^{2}}}+8\,{\frac{1/7\, \left ( bx+a \right ) ^{7/2}-2/5\, \left ( bx+a \right ) ^{5/2}a+1/3\,{a}^{2} \left ( bx+a \right ) ^{3/2}}{ \left ( a-c \right ) ^{3}{b}^{2}}}-8\,{\frac{1/7\, \left ( bx+c \right ) ^{7/2}-2/5\, \left ( bx+c \right ) ^{5/2}c+1/3\,{c}^{2} \left ( bx+c \right ) ^{3/2}}{ \left ( a-c \right ) ^{3}{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (\sqrt{b x + a} + \sqrt{b x + c}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(b*x + a) + sqrt(b*x + c))^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275982, size = 215, normalized size = 0.82 \[ \frac{2 \,{\left ({\left (20 \, b^{3} x^{3} + 6 \, a^{3} - 14 \, a^{2} c +{\left (11 \, a b^{2} + 21 \, b^{2} c\right )} x^{2} -{\left (3 \, a^{2} b - 7 \, a b c\right )} x\right )} \sqrt{b x + a} -{\left (20 \, b^{3} x^{3} - 14 \, a c^{2} + 6 \, c^{3} +{\left (21 \, a b^{2} + 11 \, b^{2} c\right )} x^{2} +{\left (7 \, a b c - 3 \, b c^{2}\right )} x\right )} \sqrt{b x + c}\right )}}{35 \,{\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(b*x + a) + sqrt(b*x + c))^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.27523, size = 942, normalized size = 3.61 \[ \begin{cases} \frac{12 a^{2}}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{54 a b x}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{44 a c}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{36 a \sqrt{a + b x} \sqrt{b x + c}}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{40 b^{2} x^{2}}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{54 b c x}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{30 b x \sqrt{a + b x} \sqrt{b x + c}}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{12 c^{2}}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} + \frac{36 c \sqrt{a + b x} \sqrt{b x + c}}{35 a b^{2} \sqrt{a + b x} + 105 a b^{2} \sqrt{b x + c} + 140 b^{3} x \sqrt{a + b x} + 140 b^{3} x \sqrt{b x + c} + 105 b^{2} c \sqrt{a + b x} + 35 b^{2} c \sqrt{b x + c}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 \left (\sqrt{a} + \sqrt{c}\right )^{3}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x+a)**(1/2)+(b*x+c)**(1/2))**3,x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(sqrt(b*x + a) + sqrt(b*x + c))^3,x, algorithm="giac")
[Out]