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

Optimal. Leaf size=92 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2} \]

[Out]

(5*a^2*Sqrt[x]*Sqrt[a + b*x])/8 + (5*a*Sqrt[x]*(a + b*x)^(3/2))/12 + (Sqrt[x]*(a
+ b*x)^(5/2))/3 + (5*a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*Sqrt[b])

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Rubi [A]  time = 0.0680556, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/Sqrt[x],x]

[Out]

(5*a^2*Sqrt[x]*Sqrt[a + b*x])/8 + (5*a*Sqrt[x]*(a + b*x)^(3/2))/12 + (Sqrt[x]*(a
+ b*x)^(5/2))/3 + (5*a^3*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(8*Sqrt[b])

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Rubi in Sympy [A]  time = 10.1825, size = 85, normalized size = 0.92 \[ \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{8 \sqrt{b}} + \frac{5 a^{2} \sqrt{x} \sqrt{a + b x}}{8} + \frac{5 a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{12} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{3} \]

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**3*atanh(sqrt(a + b*x)/(sqrt(b)*sqrt(x)))/(8*sqrt(b)) + 5*a**2*sqrt(x)*sqrt(
a + b*x)/8 + 5*a*sqrt(x)*(a + b*x)**(3/2)/12 + sqrt(x)*(a + b*x)**(5/2)/3

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Mathematica [A]  time = 0.0652762, size = 74, normalized size = 0.8 \[ \frac{5 a^3 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{8 \sqrt{b}}+\frac{1}{24} \sqrt{x} \sqrt{a+b x} \left (33 a^2+26 a b x+8 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*Sqrt[a + b*x]*(33*a^2 + 26*a*b*x + 8*b^2*x^2))/24 + (5*a^3*Log[b*Sqrt[x
] + Sqrt[b]*Sqrt[a + b*x]])/(8*Sqrt[b])

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Maple [A]  time = 0.009, size = 93, normalized size = 1. \[{\frac{1}{3} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5\,a}{12} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5\,{a}^{2}}{8}\sqrt{x}\sqrt{bx+a}}+{\frac{5\,{a}^{3}}{16}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/x^(1/2),x)

[Out]

1/3*(b*x+a)^(5/2)*x^(1/2)+5/12*a*(b*x+a)^(3/2)*x^(1/2)+5/8*a^2*x^(1/2)*(b*x+a)^(
1/2)+5/16*a^3*(x*(b*x+a))^(1/2)/(b*x+a)^(1/2)/x^(1/2)*ln((1/2*a+b*x)/b^(1/2)+(b*
x^2+a*x)^(1/2))/b^(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 = 0.224611, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 26 \, a b x + 33 \, a^{2}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{48 \, \sqrt{b}}, \frac{15 \, a^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (8 \, b^{2} x^{2} + 26 \, a b x + 33 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*a^3*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(8*b^2*x^
2 + 26*a*b*x + 33*a^2)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/sqrt(b), 1/24*(15*a^3*arct
an(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (8*b^2*x^2 + 26*a*b*x + 33*a^2)*sqrt(b*
x + a)*sqrt(-b)*sqrt(x))/sqrt(-b)]

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Sympy [A]  time = 61.0818, size = 102, normalized size = 1.11 \[ \frac{11 a^{\frac{5}{2}} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{8} + \frac{13 a^{\frac{3}{2}} b x^{\frac{3}{2}} \sqrt{1 + \frac{b x}{a}}}{12} + \frac{\sqrt{a} b^{2} x^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}}}{3} + \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11*a**(5/2)*sqrt(x)*sqrt(1 + b*x/a)/8 + 13*a**(3/2)*b*x**(3/2)*sqrt(1 + b*x/a)/1
2 + sqrt(a)*b**2*x**(5/2)*sqrt(1 + b*x/a)/3 + 5*a**3*asinh(sqrt(b)*sqrt(x)/sqrt(
a))/(8*sqrt(b))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError