3.479 \(\int x^{3/2} (a+b x)^{3/2} (A+B x) \, dx\)

Optimal. Leaf size=192 \[ \frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{7/2}}-\frac{3 a^3 \sqrt{x} \sqrt{a+b x} (2 A b-a B)}{128 b^3}+\frac{a^2 x^{3/2} \sqrt{a+b x} (2 A b-a B)}{64 b^2}+\frac{a x^{5/2} \sqrt{a+b x} (2 A b-a B)}{16 b}+\frac{x^{5/2} (a+b x)^{3/2} (2 A b-a B)}{8 b}+\frac{B x^{5/2} (a+b x)^{5/2}}{5 b} \]

[Out]

(-3*a^3*(2*A*b - a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^3) + (a^2*(2*A*b - a*B)*x^(3
/2)*Sqrt[a + b*x])/(64*b^2) + (a*(2*A*b - a*B)*x^(5/2)*Sqrt[a + b*x])/(16*b) + (
(2*A*b - a*B)*x^(5/2)*(a + b*x)^(3/2))/(8*b) + (B*x^(5/2)*(a + b*x)^(5/2))/(5*b)
+ (3*a^4*(2*A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(128*b^(7/2))

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Rubi [A]  time = 0.227886, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{7/2}}-\frac{3 a^3 \sqrt{x} \sqrt{a+b x} (2 A b-a B)}{128 b^3}+\frac{a^2 x^{3/2} \sqrt{a+b x} (2 A b-a B)}{64 b^2}+\frac{a x^{5/2} \sqrt{a+b x} (2 A b-a B)}{16 b}+\frac{x^{5/2} (a+b x)^{3/2} (2 A b-a B)}{8 b}+\frac{B x^{5/2} (a+b x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a^3*(2*A*b - a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^3) + (a^2*(2*A*b - a*B)*x^(3
/2)*Sqrt[a + b*x])/(64*b^2) + (a*(2*A*b - a*B)*x^(5/2)*Sqrt[a + b*x])/(16*b) + (
(2*A*b - a*B)*x^(5/2)*(a + b*x)^(3/2))/(8*b) + (B*x^(5/2)*(a + b*x)^(5/2))/(5*b)
+ (3*a^4*(2*A*b - a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(128*b^(7/2))

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Rubi in Sympy [A]  time = 22.3747, size = 177, normalized size = 0.92 \[ \frac{B x^{\frac{5}{2}} \left (a + b x\right )^{\frac{5}{2}}}{5 b} + \frac{3 a^{4} \left (A b - \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{64 b^{\frac{7}{2}}} + \frac{3 a^{3} \sqrt{x} \sqrt{a + b x} \left (A b - \frac{B a}{2}\right )}{64 b^{3}} + \frac{a^{2} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (A b - \frac{B a}{2}\right )}{32 b^{3}} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{5}{2}} \left (2 A b - B a\right )}{16 b^{3}} + \frac{x^{\frac{3}{2}} \left (a + b x\right )^{\frac{5}{2}} \left (A b - \frac{B a}{2}\right )}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

B*x**(5/2)*(a + b*x)**(5/2)/(5*b) + 3*a**4*(A*b - B*a/2)*atanh(sqrt(b)*sqrt(x)/s
qrt(a + b*x))/(64*b**(7/2)) + 3*a**3*sqrt(x)*sqrt(a + b*x)*(A*b - B*a/2)/(64*b**
3) + a**2*sqrt(x)*(a + b*x)**(3/2)*(A*b - B*a/2)/(32*b**3) - a*sqrt(x)*(a + b*x)
**(5/2)*(2*A*b - B*a)/(16*b**3) + x**(3/2)*(a + b*x)**(5/2)*(A*b - B*a/2)/(4*b**
2)

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Mathematica [A]  time = 0.155002, size = 137, normalized size = 0.71 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^4 B-10 a^3 b (3 A+B x)+4 a^2 b^2 x (5 A+2 B x)+16 a b^3 x^2 (15 A+11 B x)+32 b^4 x^3 (5 A+4 B x)\right )-15 a^4 (a B-2 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{640 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^4*B - 10*a^3*b*(3*A + B*x) + 4*a^2*b^2*x*(5
*A + 2*B*x) + 32*b^4*x^3*(5*A + 4*B*x) + 16*a*b^3*x^2*(15*A + 11*B*x)) - 15*a^4*
(-2*A*b + a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(640*b^(7/2))

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Maple [A]  time = 0.017, size = 260, normalized size = 1.4 \[{\frac{1}{1280}\sqrt{x}\sqrt{bx+a} \left ( 256\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+320\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+352\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+480\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+16\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+40\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-20\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+30\,A{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-60\,A{a}^{3}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-15\,B{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +30\,B{a}^{4}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280*x^(1/2)*(b*x+a)^(1/2)/b^(7/2)*(256*B*x^4*b^(9/2)*(x*(b*x+a))^(1/2)+320*A*
x^3*b^(9/2)*(x*(b*x+a))^(1/2)+352*B*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)+480*A*x^2*a*
b^(7/2)*(x*(b*x+a))^(1/2)+16*B*x^2*a^2*b^(5/2)*(x*(b*x+a))^(1/2)+40*A*a^2*(x*(b*
x+a))^(1/2)*x*b^(5/2)-20*B*a^3*(x*(b*x+a))^(1/2)*x*b^(3/2)+30*A*a^4*ln(1/2*(2*(x
*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-60*A*a^3*(x*(b*x+a))^(1/2)*b^(3/2)-1
5*B*a^5*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+30*B*a^4*(x*(b*x+a
))^(1/2)*b^(1/2))/(x*(b*x+a))^(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*x + A)*(b*x + a)^(3/2)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245723, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (128 \, B b^{4} x^{4} + 15 \, B a^{4} - 30 \, A a^{3} b + 16 \,{\left (11 \, B a b^{3} + 10 \, A b^{4}\right )} x^{3} + 8 \,{\left (B a^{2} b^{2} + 30 \, A a b^{3}\right )} x^{2} - 10 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 15 \,{\left (B a^{5} - 2 \, A a^{4} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{1280 \, b^{\frac{7}{2}}}, \frac{{\left (128 \, B b^{4} x^{4} + 15 \, B a^{4} - 30 \, A a^{3} b + 16 \,{\left (11 \, B a b^{3} + 10 \, A b^{4}\right )} x^{3} + 8 \,{\left (B a^{2} b^{2} + 30 \, A a b^{3}\right )} x^{2} - 10 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 15 \,{\left (B a^{5} - 2 \, A a^{4} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{640 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1280*(2*(128*B*b^4*x^4 + 15*B*a^4 - 30*A*a^3*b + 16*(11*B*a*b^3 + 10*A*b^4)*x
^3 + 8*(B*a^2*b^2 + 30*A*a*b^3)*x^2 - 10*(B*a^3*b - 2*A*a^2*b^2)*x)*sqrt(b*x + a
)*sqrt(b)*sqrt(x) - 15*(B*a^5 - 2*A*a^4*b)*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*
x + a)*sqrt(b)))/b^(7/2), 1/640*((128*B*b^4*x^4 + 15*B*a^4 - 30*A*a^3*b + 16*(11
*B*a*b^3 + 10*A*b^4)*x^3 + 8*(B*a^2*b^2 + 30*A*a*b^3)*x^2 - 10*(B*a^3*b - 2*A*a^
2*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) - 15*(B*a^5 - 2*A*a^4*b)*arctan(sqrt(b*
x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^3)]

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Sympy [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**(3/2)*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

Timed 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((B*x + A)*(b*x + a)^(3/2)*x^(3/2),x, algorithm="giac")

[Out]

Timed out