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

Optimal. Leaf size=159 \[ \frac{a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{7/2}}-\frac{a^2 \sqrt{x} \sqrt{a+b x} (8 A b-5 a B)}{64 b^3}+\frac{a x^{3/2} \sqrt{a+b x} (8 A b-5 a B)}{96 b^2}+\frac{x^{5/2} \sqrt{a+b x} (8 A b-5 a B)}{24 b}+\frac{B x^{5/2} (a+b x)^{3/2}}{4 b} \]

[Out]

-(a^2*(8*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^3) + (a*(8*A*b - 5*a*B)*x^(3/
2)*Sqrt[a + b*x])/(96*b^2) + ((8*A*b - 5*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b) + (B
*x^(5/2)*(a + b*x)^(3/2))/(4*b) + (a^3*(8*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])
/Sqrt[a + b*x]])/(64*b^(7/2))

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Rubi [A]  time = 0.180836, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{7/2}}-\frac{a^2 \sqrt{x} \sqrt{a+b x} (8 A b-5 a B)}{64 b^3}+\frac{a x^{3/2} \sqrt{a+b x} (8 A b-5 a B)}{96 b^2}+\frac{x^{5/2} \sqrt{a+b x} (8 A b-5 a B)}{24 b}+\frac{B x^{5/2} (a+b x)^{3/2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*(8*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^3) + (a*(8*A*b - 5*a*B)*x^(3/
2)*Sqrt[a + b*x])/(96*b^2) + ((8*A*b - 5*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b) + (B
*x^(5/2)*(a + b*x)^(3/2))/(4*b) + (a^3*(8*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])
/Sqrt[a + b*x]])/(64*b^(7/2))

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Rubi in Sympy [A]  time = 16.7685, size = 150, normalized size = 0.94 \[ \frac{B x^{\frac{5}{2}} \left (a + b x\right )^{\frac{3}{2}}}{4 b} + \frac{a^{3} \left (8 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{64 b^{\frac{7}{2}}} + \frac{a^{2} \sqrt{x} \sqrt{a + b x} \left (8 A b - 5 B a\right )}{64 b^{3}} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - 5 B a\right )}{32 b^{3}} + \frac{x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}} \left (8 A b - 5 B a\right )}{24 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

B*x**(5/2)*(a + b*x)**(3/2)/(4*b) + a**3*(8*A*b - 5*B*a)*atanh(sqrt(b)*sqrt(x)/s
qrt(a + b*x))/(64*b**(7/2)) + a**2*sqrt(x)*sqrt(a + b*x)*(8*A*b - 5*B*a)/(64*b**
3) - a*sqrt(x)*(a + b*x)**(3/2)*(8*A*b - 5*B*a)/(32*b**3) + x**(3/2)*(a + b*x)**
(3/2)*(8*A*b - 5*B*a)/(24*b**2)

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Mathematica [A]  time = 0.126127, size = 119, normalized size = 0.75 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^3 B-2 a^2 b (12 A+5 B x)+8 a b^2 x (2 A+B x)+16 b^3 x^2 (4 A+3 B x)\right )-3 a^3 (5 a B-8 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{192 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^3*B + 8*a*b^2*x*(2*A + B*x) + 16*b^3*x^2*(4
*A + 3*B*x) - 2*a^2*b*(12*A + 5*B*x)) - 3*a^3*(-8*A*b + 5*a*B)*Log[b*Sqrt[x] + S
qrt[b]*Sqrt[a + b*x]])/(192*b^(7/2))

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Maple [A]  time = 0.027, size = 218, normalized size = 1.4 \[{\frac{1}{384}\sqrt{x}\sqrt{bx+a} \left ( 96\,B{x}^{3}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+128\,A{x}^{2}{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+16\,B{x}^{2}a{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+32\,Aax\sqrt{x \left ( bx+a \right ) }{b}^{5/2}-20\,B{a}^{2}x\sqrt{x \left ( bx+a \right ) }{b}^{3/2}+24\,A{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-48\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }{b}^{3/2}-15\,B{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +30\,B{a}^{3}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)*(b*x+a)^(1/2),x)

[Out]

1/384*x^(1/2)*(b*x+a)^(1/2)/b^(7/2)*(96*B*x^3*b^(7/2)*(x*(b*x+a))^(1/2)+128*A*x^
2*b^(7/2)*(x*(b*x+a))^(1/2)+16*B*x^2*a*b^(5/2)*(x*(b*x+a))^(1/2)+32*A*a*x*(x*(b*
x+a))^(1/2)*b^(5/2)-20*B*a^2*x*(x*(b*x+a))^(1/2)*b^(3/2)+24*A*a^3*ln(1/2*(2*(x*(
b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-48*A*a^2*(x*(b*x+a))^(1/2)*b^(3/2)-15*
B*a^4*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+30*B*a^3*(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)*sqrt(b*x + a)*x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247928, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B b^{3} x^{3} + 15 \, B a^{3} - 24 \, A a^{2} b + 8 \,{\left (B a b^{2} + 8 \, A b^{3}\right )} x^{2} - 2 \,{\left (5 \, B a^{2} b - 8 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 3 \,{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{384 \, b^{\frac{7}{2}}}, \frac{{\left (48 \, B b^{3} x^{3} + 15 \, B a^{3} - 24 \, A a^{2} b + 8 \,{\left (B a b^{2} + 8 \, A b^{3}\right )} x^{2} - 2 \,{\left (5 \, B a^{2} b - 8 \, A a b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 3 \,{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{192 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(2*(48*B*b^3*x^3 + 15*B*a^3 - 24*A*a^2*b + 8*(B*a*b^2 + 8*A*b^3)*x^2 - 2*
(5*B*a^2*b - 8*A*a*b^2)*x)*sqrt(b*x + a)*sqrt(b)*sqrt(x) - 3*(5*B*a^4 - 8*A*a^3*
b)*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)))/b^(7/2), 1/192*((48*B*b
^3*x^3 + 15*B*a^3 - 24*A*a^2*b + 8*(B*a*b^2 + 8*A*b^3)*x^2 - 2*(5*B*a^2*b - 8*A*
a*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) - 3*(5*B*a^4 - 8*A*a^3*b)*arctan(sqrt(b
*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 81.5794, size = 1527, normalized size = 9.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(B*x+A)*(b*x+a)**(1/2),x)

[Out]

-2*A*a*Piecewise((a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(b*x/a)) - 3*sqrt(a)*(a
+ b*x)**(3/2)/(8*sqrt(b)*sqrt(b*x/a)) - a**2*acosh(sqrt(a + b*x)/sqrt(a))/(8*sqr
t(b)) + (a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b)*sqrt(b*x/a)), Abs(1 + b*x/a) > 1), (
-I*a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(-b*x/a)) + 3*I*sqrt(a)*(a + b*x)**(3/2
)/(8*sqrt(b)*sqrt(-b*x/a)) + I*a**2*asin(sqrt(a + b*x)/sqrt(a))/(8*sqrt(b)) - I*
(a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**2 + 2*A*Piecewise((
a**(5/2)*sqrt(a + b*x)/(16*sqrt(b)*sqrt(b*x/a)) - a**(3/2)*(a + b*x)**(3/2)/(48*
sqrt(b)*sqrt(b*x/a)) - 5*sqrt(a)*(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(b*x/a)) - a**
3*acosh(sqrt(a + b*x)/sqrt(a))/(16*sqrt(b)) + (a + b*x)**(7/2)/(6*sqrt(a)*sqrt(b
)*sqrt(b*x/a)), Abs(1 + b*x/a) > 1), (-I*a**(5/2)*sqrt(a + b*x)/(16*sqrt(b)*sqrt
(-b*x/a)) + I*a**(3/2)*(a + b*x)**(3/2)/(48*sqrt(b)*sqrt(-b*x/a)) + 5*I*sqrt(a)*
(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(-b*x/a)) + I*a**3*asin(sqrt(a + b*x)/sqrt(a))/
(16*sqrt(b)) - I*(a + b*x)**(7/2)/(6*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**2
+ 2*B*a**2*Piecewise((a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(b*x/a)) - 3*sqrt(a)
*(a + b*x)**(3/2)/(8*sqrt(b)*sqrt(b*x/a)) - a**2*acosh(sqrt(a + b*x)/sqrt(a))/(8
*sqrt(b)) + (a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b)*sqrt(b*x/a)), Abs(1 + b*x/a) > 1
), (-I*a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(-b*x/a)) + 3*I*sqrt(a)*(a + b*x)**
(3/2)/(8*sqrt(b)*sqrt(-b*x/a)) + I*a**2*asin(sqrt(a + b*x)/sqrt(a))/(8*sqrt(b))
- I*(a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**3 - 4*B*a*Piece
wise((a**(5/2)*sqrt(a + b*x)/(16*sqrt(b)*sqrt(b*x/a)) - a**(3/2)*(a + b*x)**(3/2
)/(48*sqrt(b)*sqrt(b*x/a)) - 5*sqrt(a)*(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(b*x/a))
- a**3*acosh(sqrt(a + b*x)/sqrt(a))/(16*sqrt(b)) + (a + b*x)**(7/2)/(6*sqrt(a)*
sqrt(b)*sqrt(b*x/a)), Abs(1 + b*x/a) > 1), (-I*a**(5/2)*sqrt(a + b*x)/(16*sqrt(b
)*sqrt(-b*x/a)) + I*a**(3/2)*(a + b*x)**(3/2)/(48*sqrt(b)*sqrt(-b*x/a)) + 5*I*sq
rt(a)*(a + b*x)**(5/2)/(24*sqrt(b)*sqrt(-b*x/a)) + I*a**3*asin(sqrt(a + b*x)/sqr
t(a))/(16*sqrt(b)) - I*(a + b*x)**(7/2)/(6*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))
/b**3 + 2*B*Piecewise((5*a**(7/2)*sqrt(a + b*x)/(128*sqrt(b)*sqrt(b*x/a)) - 5*a*
*(5/2)*(a + b*x)**(3/2)/(384*sqrt(b)*sqrt(b*x/a)) - a**(3/2)*(a + b*x)**(5/2)/(1
92*sqrt(b)*sqrt(b*x/a)) - 7*sqrt(a)*(a + b*x)**(7/2)/(48*sqrt(b)*sqrt(b*x/a)) -
5*a**4*acosh(sqrt(a + b*x)/sqrt(a))/(128*sqrt(b)) + (a + b*x)**(9/2)/(8*sqrt(a)*
sqrt(b)*sqrt(b*x/a)), Abs(1 + b*x/a) > 1), (-5*I*a**(7/2)*sqrt(a + b*x)/(128*sqr
t(b)*sqrt(-b*x/a)) + 5*I*a**(5/2)*(a + b*x)**(3/2)/(384*sqrt(b)*sqrt(-b*x/a)) +
I*a**(3/2)*(a + b*x)**(5/2)/(192*sqrt(b)*sqrt(-b*x/a)) + 7*I*sqrt(a)*(a + b*x)**
(7/2)/(48*sqrt(b)*sqrt(-b*x/a)) + 5*I*a**4*asin(sqrt(a + b*x)/sqrt(a))/(128*sqrt
(b)) - I*(a + b*x)**(9/2)/(8*sqrt(a)*sqrt(b)*sqrt(-b*x/a)), True))/b**3

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

[Out]

Timed out