3.1146 \(\int (a+b x)^{3/2} (a c-b c x)^{3/2} \, dx\)

Optimal. Leaf size=103 \[ \frac{3 a^4 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{4 b}+\frac{3}{8} a^2 c x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2} \]

[Out]

(3*a^2*c*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/8 + (x*(a + b*x)^(3/2)*(a*c - b*c*x)
^(3/2))/4 + (3*a^4*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*c - b*c*x]])/(4
*b)

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Rubi [A]  time = 0.108795, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{3 a^4 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{4 b}+\frac{3}{8} a^2 c x \sqrt{a+b x} \sqrt{a c-b c x}+\frac{1}{4} x (a+b x)^{3/2} (a c-b c x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]

[Out]

(3*a^2*c*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/8 + (x*(a + b*x)^(3/2)*(a*c - b*c*x)
^(3/2))/4 + (3*a^4*c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a*c - b*c*x]])/(4
*b)

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Rubi in Sympy [A]  time = 18.605, size = 94, normalized size = 0.91 \[ \frac{3 a^{4} c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a c - b c x}} \right )}}{4 b} + \frac{3 a^{2} c x \sqrt{a + b x} \sqrt{a c - b c x}}{8} + \frac{x \left (a + b x\right )^{\frac{3}{2}} \left (a c - b c x\right )^{\frac{3}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2),x)

[Out]

3*a**4*c**(3/2)*atan(sqrt(c)*sqrt(a + b*x)/sqrt(a*c - b*c*x))/(4*b) + 3*a**2*c*x
*sqrt(a + b*x)*sqrt(a*c - b*c*x)/8 + x*(a + b*x)**(3/2)*(a*c - b*c*x)**(3/2)/4

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Mathematica [A]  time = 0.114805, size = 94, normalized size = 0.91 \[ \frac{(c (a-b x))^{3/2} \left (3 a^4 \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right )+b x \sqrt{a-b x} \sqrt{a+b x} \left (5 a^2-2 b^2 x^2\right )\right )}{8 b (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2),x]

[Out]

((c*(a - b*x))^(3/2)*(b*x*Sqrt[a - b*x]*Sqrt[a + b*x]*(5*a^2 - 2*b^2*x^2) + 3*a^
4*ArcTan[(b*x)/(Sqrt[a - b*x]*Sqrt[a + b*x])]))/(8*b*(a - b*x)^(3/2))

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Maple [B]  time = 0.008, size = 185, normalized size = 1.8 \[ -{\frac{1}{4\,bc} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( -bcx+ac \right ) ^{{\frac{5}{2}}}}-{\frac{a}{4\,bc}\sqrt{bx+a} \left ( -bcx+ac \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}}{8\,b} \left ( -bcx+ac \right ) ^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{3\,{a}^{3}c}{8\,b}\sqrt{bx+a}\sqrt{-bcx+ac}}+{\frac{3\,{a}^{4}{c}^{2}}{8}\sqrt{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-{b}^{2}c{x}^{2}+{a}^{2}c}}}} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{-bcx+ac}}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2),x)

[Out]

-1/4/b/c*(b*x+a)^(3/2)*(-b*c*x+a*c)^(5/2)-1/4*a/b/c*(b*x+a)^(1/2)*(-b*c*x+a*c)^(
5/2)+1/8*a^2/b*(-b*c*x+a*c)^(3/2)*(b*x+a)^(1/2)+3/8*a^3*c/b*(-b*c*x+a*c)^(1/2)*(
b*x+a)^(1/2)+3/8*a^4*c^2*((b*x+a)*(-b*c*x+a*c))^(1/2)/(-b*c*x+a*c)^(1/2)/(b*x+a)
^(1/2)/(b^2*c)^(1/2)*arctan((b^2*c)^(1/2)*x/(-b^2*c*x^2+a^2*c)^(1/2))

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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*c*x + a*c)^(3/2)*(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228616, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{4} \sqrt{-c} c \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) - 2 \,{\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{16 \, b}, \frac{3 \, a^{4} c^{\frac{3}{2}} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right ) -{\left (2 \, b^{3} c x^{3} - 5 \, a^{2} b c x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{8 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(3*a^4*sqrt(-c)*c*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*s
qrt(-c)*x - a^2*c) - 2*(2*b^3*c*x^3 - 5*a^2*b*c*x)*sqrt(-b*c*x + a*c)*sqrt(b*x +
a))/b, 1/8*(3*a^4*c^(3/2)*arctan(b*sqrt(c)*x/(sqrt(-b*c*x + a*c)*sqrt(b*x + a))
) - (2*b^3*c*x^3 - 5*a^2*b*c*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- c \left (- a + b x\right )\right )^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(-b*c*x+a*c)**(3/2),x)

[Out]

Integral((-c*(-a + b*x))**(3/2)*(a + b*x)**(3/2), x)

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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((-b*c*x + a*c)^(3/2)*(b*x + a)^(3/2),x, algorithm="giac")

[Out]

Timed out