3.547 \(\int \sqrt{x} (a+b x)^{5/2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^3 \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a+b x}+\frac{5}{24} a x^{3/2} (a+b x)^{3/2}+\frac{1}{4} x^{3/2} (a+b x)^{5/2} \]

[Out]

(5*a^3*Sqrt[x]*Sqrt[a + b*x])/(64*b) + (5*a^2*x^(3/2)*Sqrt[a + b*x])/32 + (5*a*x
^(3/2)*(a + b*x)^(3/2))/24 + (x^(3/2)*(a + b*x)^(5/2))/4 - (5*a^4*ArcTanh[(Sqrt[
b]*Sqrt[x])/Sqrt[a + b*x]])/(64*b^(3/2))

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Rubi [A]  time = 0.0909257, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^3 \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a+b x}+\frac{5}{24} a x^{3/2} (a+b x)^{3/2}+\frac{1}{4} x^{3/2} (a+b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + b*x)^(5/2),x]

[Out]

(5*a^3*Sqrt[x]*Sqrt[a + b*x])/(64*b) + (5*a^2*x^(3/2)*Sqrt[a + b*x])/32 + (5*a*x
^(3/2)*(a + b*x)^(3/2))/24 + (x^(3/2)*(a + b*x)^(5/2))/4 - (5*a^4*ArcTanh[(Sqrt[
b]*Sqrt[x])/Sqrt[a + b*x]])/(64*b^(3/2))

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Rubi in Sympy [A]  time = 14.1474, size = 110, normalized size = 0.95 \[ - \frac{5 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{64 b^{\frac{3}{2}}} - \frac{5 a^{3} \sqrt{x} \sqrt{a + b x}}{64 b} - \frac{5 a^{2} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{96 b} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{24 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{7}{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*x**(1/2),x)

[Out]

-5*a**4*atanh(sqrt(b)*sqrt(x)/sqrt(a + b*x))/(64*b**(3/2)) - 5*a**3*sqrt(x)*sqrt
(a + b*x)/(64*b) - 5*a**2*sqrt(x)*(a + b*x)**(3/2)/(96*b) - a*sqrt(x)*(a + b*x)*
*(5/2)/(24*b) + sqrt(x)*(a + b*x)**(7/2)/(4*b)

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Mathematica [A]  time = 0.0629352, size = 89, normalized size = 0.77 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^3+118 a^2 b x+136 a b^2 x^2+48 b^3 x^3\right )-15 a^4 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{192 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(a + b*x)^(5/2),x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(15*a^3 + 118*a^2*b*x + 136*a*b^2*x^2 + 48*b^3*x^
3) - 15*a^4*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(192*b^(3/2))

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Maple [A]  time = 0.008, size = 111, normalized size = 1. \[{\frac{1}{4}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a}{24}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}}{32}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{3}}{64\,b}\sqrt{x}\sqrt{bx+a}}-{\frac{5\,{a}^{4}}{128}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*x^(1/2),x)

[Out]

1/4*x^(3/2)*(b*x+a)^(5/2)+5/24*a*x^(3/2)*(b*x+a)^(3/2)+5/32*a^2*x^(3/2)*(b*x+a)^
(1/2)+5/64*a^3*x^(1/2)*(b*x+a)^(1/2)/b-5/128*a^4/b^(3/2)*(x*(b*x+a))^(1/2)/x^(1/
2)/(b*x+a)^(1/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(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)^(5/2)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224735, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{384 \, b^{\frac{3}{2}}}, -\frac{15 \, a^{4} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{192 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*sqrt(x),x, algorithm="fricas")

[Out]

[1/384*(15*a^4*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(48*b^3
*x^3 + 136*a*b^2*x^2 + 118*a^2*b*x + 15*a^3)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/b^(3
/2), -1/192*(15*a^4*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (48*b^3*x^3 + 1
36*a*b^2*x^2 + 118*a^2*b*x + 15*a^3)*sqrt(b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b
)]

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Sympy [A]  time = 101.738, size = 155, normalized size = 1.34 \[ \frac{5 a^{\frac{7}{2}} \sqrt{x}}{64 b \sqrt{1 + \frac{b x}{a}}} + \frac{133 a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 \sqrt{1 + \frac{b x}{a}}} + \frac{127 a^{\frac{3}{2}} b x^{\frac{5}{2}}}{96 \sqrt{1 + \frac{b x}{a}}} + \frac{23 \sqrt{a} b^{2} x^{\frac{7}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} - \frac{5 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{b^{3} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*x**(1/2),x)

[Out]

5*a**(7/2)*sqrt(x)/(64*b*sqrt(1 + b*x/a)) + 133*a**(5/2)*x**(3/2)/(192*sqrt(1 +
b*x/a)) + 127*a**(3/2)*b*x**(5/2)/(96*sqrt(1 + b*x/a)) + 23*sqrt(a)*b**2*x**(7/2
)/(24*sqrt(1 + b*x/a)) - 5*a**4*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(64*b**(3/2)) + b
**3*x**(9/2)/(4*sqrt(a)*sqrt(1 + b*x/a))

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GIAC/XCAS [A]  time = 24.6269, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*sqrt(x),x, algorithm="giac")

[Out]

sage0*x