3.284 \(\int \frac {(a+b x^2)^3}{\sqrt {x}} \, dx\)

Optimal. Leaf size=49 \[ 2 a^3 \sqrt {x}+\frac {6}{5} a^2 b x^{5/2}+\frac {2}{3} a b^2 x^{9/2}+\frac {2}{13} b^3 x^{13/2} \]

[Out]

6/5*a^2*b*x^(5/2)+2/3*a*b^2*x^(9/2)+2/13*b^3*x^(13/2)+2*a^3*x^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \frac {6}{5} a^2 b x^{5/2}+2 a^3 \sqrt {x}+\frac {2}{3} a b^2 x^{9/2}+\frac {2}{13} b^3 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/Sqrt[x],x]

[Out]

2*a^3*Sqrt[x] + (6*a^2*b*x^(5/2))/5 + (2*a*b^2*x^(9/2))/3 + (2*b^3*x^(13/2))/13

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^3}{\sqrt {x}} \, dx &=\int \left (\frac {a^3}{\sqrt {x}}+3 a^2 b x^{3/2}+3 a b^2 x^{7/2}+b^3 x^{11/2}\right ) \, dx\\ &=2 a^3 \sqrt {x}+\frac {6}{5} a^2 b x^{5/2}+\frac {2}{3} a b^2 x^{9/2}+\frac {2}{13} b^3 x^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.84 \[ \frac {2}{195} \sqrt {x} \left (195 a^3+117 a^2 b x^2+65 a b^2 x^4+15 b^3 x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(195*a^3 + 117*a^2*b*x^2 + 65*a*b^2*x^4 + 15*b^3*x^6))/195

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fricas [A]  time = 0.85, size = 37, normalized size = 0.76 \[ \frac {2}{195} \, {\left (15 \, b^{3} x^{6} + 65 \, a b^{2} x^{4} + 117 \, a^{2} b x^{2} + 195 \, a^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/195*(15*b^3*x^6 + 65*a*b^2*x^4 + 117*a^2*b*x^2 + 195*a^3)*sqrt(x)

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giac [A]  time = 0.59, size = 35, normalized size = 0.71 \[ \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + 2 \, a^{3} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^(1/2),x, algorithm="giac")

[Out]

2/13*b^3*x^(13/2) + 2/3*a*b^2*x^(9/2) + 6/5*a^2*b*x^(5/2) + 2*a^3*sqrt(x)

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maple [A]  time = 0.01, size = 38, normalized size = 0.78 \[ \frac {2 \left (15 b^{3} x^{6}+65 a \,b^{2} x^{4}+117 a^{2} b \,x^{2}+195 a^{3}\right ) \sqrt {x}}{195} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^(1/2),x)

[Out]

2/195*x^(1/2)*(15*b^3*x^6+65*a*b^2*x^4+117*a^2*b*x^2+195*a^3)

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maxima [A]  time = 1.34, size = 35, normalized size = 0.71 \[ \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, a b^{2} x^{\frac {9}{2}} + \frac {6}{5} \, a^{2} b x^{\frac {5}{2}} + 2 \, a^{3} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/13*b^3*x^(13/2) + 2/3*a*b^2*x^(9/2) + 6/5*a^2*b*x^(5/2) + 2*a^3*sqrt(x)

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mupad [B]  time = 0.04, size = 35, normalized size = 0.71 \[ 2\,a^3\,\sqrt {x}+\frac {2\,b^3\,x^{13/2}}{13}+\frac {6\,a^2\,b\,x^{5/2}}{5}+\frac {2\,a\,b^2\,x^{9/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^(1/2),x)

[Out]

2*a^3*x^(1/2) + (2*b^3*x^(13/2))/13 + (6*a^2*b*x^(5/2))/5 + (2*a*b^2*x^(9/2))/3

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sympy [A]  time = 2.10, size = 48, normalized size = 0.98 \[ 2 a^{3} \sqrt {x} + \frac {6 a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 b^{3} x^{\frac {13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**(1/2),x)

[Out]

2*a**3*sqrt(x) + 6*a**2*b*x**(5/2)/5 + 2*a*b**2*x**(9/2)/3 + 2*b**3*x**(13/2)/13

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