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

Optimal. Leaf size=126 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}-\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-5 a B)}{8 b^3}+\frac{x^{3/2} \sqrt{a+b x} (6 A b-5 a B)}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b} \]

[Out]

-(a*(6*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(8*b^3) + ((6*A*b - 5*a*B)*x^(3/2)*Sq
rt[a + b*x])/(12*b^2) + (B*x^(5/2)*Sqrt[a + b*x])/(3*b) + (a^2*(6*A*b - 5*a*B)*A
rcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(7/2))

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Rubi [A]  time = 0.142528, 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 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}-\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-5 a B)}{8 b^3}+\frac{x^{3/2} \sqrt{a+b x} (6 A b-5 a B)}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(6*A*b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(8*b^3) + ((6*A*b - 5*a*B)*x^(3/2)*Sq
rt[a + b*x])/(12*b^2) + (B*x^(5/2)*Sqrt[a + b*x])/(3*b) + (a^2*(6*A*b - 5*a*B)*A
rcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*b^(7/2))

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Rubi in Sympy [A]  time = 12.681, size = 117, normalized size = 0.93 \[ \frac{B x^{\frac{5}{2}} \sqrt{a + b x}}{3 b} + \frac{a^{2} \left (6 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{8 b^{\frac{7}{2}}} - \frac{a \sqrt{x} \sqrt{a + b x} \left (6 A b - 5 B a\right )}{8 b^{3}} + \frac{x^{\frac{3}{2}} \sqrt{a + b x} \left (6 A b - 5 B a\right )}{12 b^{2}} \]

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)*sqrt(a + b*x)/(3*b) + a**2*(6*A*b - 5*B*a)*atanh(sqrt(b)*sqrt(x)/sqrt
(a + b*x))/(8*b**(7/2)) - a*sqrt(x)*sqrt(a + b*x)*(6*A*b - 5*B*a)/(8*b**3) + x**
(3/2)*sqrt(a + b*x)*(6*A*b - 5*B*a)/(12*b**2)

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Mathematica [A]  time = 0.111185, size = 101, normalized size = 0.8 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^2 B-2 a b (9 A+5 B x)+4 b^2 x (3 A+2 B x)\right )-3 a^2 (5 a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^2*B + 4*b^2*x*(3*A + 2*B*x) - 2*a*b*(9*A +
5*B*x)) - 3*a^2*(-6*A*b + 5*a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(24*b^(
7/2))

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Maple [A]  time = 0.022, size = 176, normalized size = 1.4 \[{\frac{1}{48}\sqrt{x}\sqrt{bx+a} \left ( 16\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+24\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-20\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+18\,A{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-36\,A\sqrt{x \left ( bx+a \right ) }a{b}^{3/2}-15\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +30\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

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]

1/48*x^(1/2)*(b*x+a)^(1/2)/b^(7/2)*(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)-20*B*a*(x*(b*x+a))^(1/2)*x*b^(3/2)+18*A*a^2*ln(1/2*(2*(x
*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*b-36*A*(x*(b*x+a))^(1/2)*a*b^(3/2)-15*
B*a^3*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))+30*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)*x^(3/2)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240406, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 18 \, A a b - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{48 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 18 \, A a b - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{24 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(8*B*b^2*x^2 + 15*B*a^2 - 18*A*a*b - 2*(5*B*a*b - 6*A*b^2)*x)*sqrt(b*x
+ a)*sqrt(b)*sqrt(x) - 3*(5*B*a^3 - 6*A*a^2*b)*log(2*sqrt(b*x + a)*b*sqrt(x) + (
2*b*x + a)*sqrt(b)))/b^(7/2), 1/24*((8*B*b^2*x^2 + 15*B*a^2 - 18*A*a*b - 2*(5*B*
a*b - 6*A*b^2)*x)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) - 3*(5*B*a^3 - 6*A*a^2*b)*arcta
n(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 99.9212, size = 245, normalized size = 1.94 \[ - \frac{3 A a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{A \sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{A x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} + \frac{5 B a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{1 + \frac{b x}{a}}} + \frac{5 B a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{B \sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{1 + \frac{b x}{a}}} - \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{B x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

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]

-3*A*a**(3/2)*sqrt(x)/(4*b**2*sqrt(1 + b*x/a)) - A*sqrt(a)*x**(3/2)/(4*b*sqrt(1
+ b*x/a)) + 3*A*a**2*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(5/2)) + A*x**(5/2)/(2
*sqrt(a)*sqrt(1 + b*x/a)) + 5*B*a**(5/2)*sqrt(x)/(8*b**3*sqrt(1 + b*x/a)) + 5*B*
a**(3/2)*x**(3/2)/(24*b**2*sqrt(1 + b*x/a)) - B*sqrt(a)*x**(5/2)/(12*b*sqrt(1 +
b*x/a)) - 5*B*a**3*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(8*b**(7/2)) + B*x**(7/2)/(3*s
qrt(a)*sqrt(1 + b*x/a))

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

[Out]

Timed out