3.3.74 \(\int x^{3/2} (a+b x^2)^2 \, dx\) [274]

Optimal. Leaf size=36 \[ \frac {2}{5} a^2 x^{5/2}+\frac {4}{9} a b x^{9/2}+\frac {2}{13} b^2 x^{13/2} \]

[Out]

2/5*a^2*x^(5/2)+4/9*a*b*x^(9/2)+2/13*b^2*x^(13/2)

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Rubi [A]
time = 0.01, antiderivative size = 36, 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 = {276} \begin {gather*} \frac {2}{5} a^2 x^{5/2}+\frac {4}{9} a b x^{9/2}+\frac {2}{13} b^2 x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x^2)^2,x]

[Out]

(2*a^2*x^(5/2))/5 + (4*a*b*x^(9/2))/9 + (2*b^2*x^(13/2))/13

Rule 276

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 x^{3/2} \left (a+b x^2\right )^2 \, dx &=\int \left (a^2 x^{3/2}+2 a b x^{7/2}+b^2 x^{11/2}\right ) \, dx\\ &=\frac {2}{5} a^2 x^{5/2}+\frac {4}{9} a b x^{9/2}+\frac {2}{13} b^2 x^{13/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.83 \begin {gather*} \frac {2}{585} x^{5/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x^2)^2,x]

[Out]

(2*x^(5/2)*(117*a^2 + 130*a*b*x^2 + 45*b^2*x^4))/585

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Maple [A]
time = 0.06, size = 25, normalized size = 0.69

method	result	size
derivativedivides	\(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {9}{2}}}{9}+\frac {2 b^{2} x^{\frac {13}{2}}}{13}\)	\(25\)
default	\(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {9}{2}}}{9}+\frac {2 b^{2} x^{\frac {13}{2}}}{13}\)	\(25\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (45 b^{2} x^{4}+130 a b \,x^{2}+117 a^{2}\right )}{585}\)	\(27\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (45 b^{2} x^{4}+130 a b \,x^{2}+117 a^{2}\right )}{585}\)	\(27\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (45 b^{2} x^{4}+130 a b \,x^{2}+117 a^{2}\right )}{585}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/5*a^2*x^(5/2)+4/9*a*b*x^(9/2)+2/13*b^2*x^(13/2)

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Maxima [A]
time = 0.27, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{13} \, b^{2} x^{\frac {13}{2}} + \frac {4}{9} \, a b x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*b^2*x^(13/2) + 4/9*a*b*x^(9/2) + 2/5*a^2*x^(5/2)

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Fricas [A]
time = 1.58, size = 29, normalized size = 0.81 \begin {gather*} \frac {2}{585} \, {\left (45 \, b^{2} x^{6} + 130 \, a b x^{4} + 117 \, a^{2} x^{2}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*(45*b^2*x^6 + 130*a*b*x^4 + 117*a^2*x^2)*sqrt(x)

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Sympy [A]
time = 0.27, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(5/2)/5 + 4*a*b*x**(9/2)/9 + 2*b**2*x**(13/2)/13

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Giac [A]
time = 1.14, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{13} \, b^{2} x^{\frac {13}{2}} + \frac {4}{9} \, a b x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*b^2*x^(13/2) + 4/9*a*b*x^(9/2) + 2/5*a^2*x^(5/2)

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Mupad [B]
time = 0.04, size = 26, normalized size = 0.72 \begin {gather*} \frac {2\,x^{5/2}\,\left (117\,a^2+130\,a\,b\,x^2+45\,b^2\,x^4\right )}{585} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(a + b*x^2)^2,x)

[Out]

(2*x^(5/2)*(117*a^2 + 45*b^2*x^4 + 130*a*b*x^2))/585

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