3.469 \(\int \sqrt{x} \sqrt{a+b x} (A+B x) \, dx\)

Optimal. Leaf size=126 \[ -\frac{a^2 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{5/2}}+\frac{a \sqrt{x} \sqrt{a+b x} (2 A b-a B)}{8 b^2}+\frac{x^{3/2} \sqrt{a+b x} (2 A b-a B)}{4 b}+\frac{B x^{3/2} (a+b x)^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.140264, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^2 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{5/2}}+\frac{a \sqrt{x} \sqrt{a+b x} (2 A b-a B)}{8 b^2}+\frac{x^{3/2} \sqrt{a+b x} (2 A b-a B)}{4 b}+\frac{B x^{3/2} (a+b x)^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[a + b*x]*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.5476, size = 112, normalized size = 0.89 \[ \frac{B x^{\frac{3}{2}} \left (a + b x\right )^{\frac{3}{2}}}{3 b} - \frac{a^{2} \left (A b - \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} - \frac{a \sqrt{x} \sqrt{a + b x} \left (2 A b - B a\right )}{8 b^{2}} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (A b - \frac{B a}{2}\right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**(3/2)*(a + b*x)**(3/2)/(3*b) - a**2*(A*b - B*a/2)*atanh(sqrt(a + b*x)/(sqrt
(b)*sqrt(x)))/(4*b**(5/2)) - a*sqrt(x)*sqrt(a + b*x)*(2*A*b - B*a)/(8*b**2) + sq
rt(x)*(a + b*x)**(3/2)*(A*b - B*a/2)/(2*b**2)

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Mathematica [A]  time = 0.102499, size = 99, normalized size = 0.79 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (-3 a^2 B+2 a b (3 A+B x)+4 b^2 x (3 A+2 B x)\right )+3 a^2 (a B-2 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[a + b*x]*(A + B*x),x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-3*a^2*B + 2*a*b*(3*A + B*x) + 4*b^2*x*(3*A + 2*
B*x)) + 3*a^2*(-2*A*b + a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(24*b^(5/2)
)

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Maple [A]  time = 0.016, size = 176, normalized size = 1.4 \[ -{\frac{1}{48}\sqrt{bx+a}\sqrt{x} \left ( -16\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }-24\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-4\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+6\,A{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-12\,A\sqrt{x \left ( bx+a \right ) }a{b}^{3/2}-3\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +6\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(b*x+a)^(1/2)*x^(1/2)/b^(5/2)*(-16*B*x^2*b^(5/2)*(x*(b*x+a))^(1/2)-24*A*(x
*(b*x+a))^(1/2)*x*b^(5/2)-4*B*a*(x*(b*x+a))^(1/2)*x*b^(3/2)+6*A*a^2*ln(1/2*(2*(x
*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-12*A*(x*(b*x+a))^(1/2)*a*b^(3/2)-3*B
*a^3*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+6*B*a^2*(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)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245407, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{2} - 3 \, B a^{2} + 6 \, A a b + 2 \,{\left (B a b + 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{48 \, b^{\frac{5}{2}}}, \frac{{\left (8 \, B b^{2} x^{2} - 3 \, B a^{2} + 6 \, A a b + 2 \,{\left (B a b + 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} + 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{24 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(8*B*b^2*x^2 - 3*B*a^2 + 6*A*a*b + 2*(B*a*b + 6*A*b^2)*x)*sqrt(b*x + a)
*sqrt(b)*sqrt(x) - 3*(B*a^3 - 2*A*a^2*b)*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x
+ a)*sqrt(b)))/b^(5/2), 1/24*((8*B*b^2*x^2 - 3*B*a^2 + 6*A*a*b + 2*(B*a*b + 6*A
*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) + 3*(B*a^3 - 2*A*a^2*b)*arctan(sqrt(b*x
+ a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 41.0882, size = 673, normalized size = 5.34 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(3/2)*sqrt(x)/(4*b*sqrt(1 + b*x/a)) + 3*A*sqrt(a)*x**(3/2)/(4*sqrt(1 + b*x/
a)) - A*a**2*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(3/2)) + A*b*x**(5/2)/(2*sqrt(
a)*sqrt(1 + b*x/a)) - 2*B*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*sqrt(b)) + (a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b)*sqrt(b*x/a)), A
bs(1 + b*x/a) > 1), (-I*a**(3/2)*sqrt(a + b*x)/(8*sqrt(b)*sqrt(-b*x/a)) + 3*I*sq
rt(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*B*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(sq
rt(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

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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)*sqrt(x),x, algorithm="giac")

[Out]

Timed out