3.281 \(\int x^{5/2} (a+b x^2)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a^2 b x^{11/2}+\frac {2}{5} a b^2 x^{15/2}+\frac {2}{19} b^3 x^{19/2} \]

[Out]

2/7*a^3*x^(7/2)+6/11*a^2*b*x^(11/2)+2/5*a*b^2*x^(15/2)+2/19*b^3*x^(19/2)

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Rubi [A]  time = 0.01, antiderivative size = 51, 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} \[ \frac {6}{11} a^2 b x^{11/2}+\frac {2}{7} a^3 x^{7/2}+\frac {2}{5} a b^2 x^{15/2}+\frac {2}{19} b^3 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*b^2*x^(15/2))/5 + (2*b^3*x^(19/2))/19

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 x^{5/2} \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 b x^{9/2}+3 a b^2 x^{13/2}+b^3 x^{17/2}\right ) \, dx\\ &=\frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a^2 b x^{11/2}+\frac {2}{5} a b^2 x^{15/2}+\frac {2}{19} b^3 x^{19/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.80 \[ \frac {2 x^{7/2} \left (1045 a^3+1995 a^2 b x^2+1463 a b^2 x^4+385 b^3 x^6\right )}{7315} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^2)^3,x]

[Out]

(2*x^(7/2)*(1045*a^3 + 1995*a^2*b*x^2 + 1463*a*b^2*x^4 + 385*b^3*x^6))/7315

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fricas [A]  time = 0.67, size = 40, normalized size = 0.78 \[ \frac {2}{7315} \, {\left (385 \, b^{3} x^{9} + 1463 \, a b^{2} x^{7} + 1995 \, a^{2} b x^{5} + 1045 \, a^{3} x^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^3,x, algorithm="fricas")

[Out]

2/7315*(385*b^3*x^9 + 1463*a*b^2*x^7 + 1995*a^2*b*x^5 + 1045*a^3*x^3)*sqrt(x)

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giac [A]  time = 0.63, size = 35, normalized size = 0.69 \[ \frac {2}{19} \, b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, a b^{2} x^{\frac {15}{2}} + \frac {6}{11} \, a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^3,x, algorithm="giac")

[Out]

2/19*b^3*x^(19/2) + 2/5*a*b^2*x^(15/2) + 6/11*a^2*b*x^(11/2) + 2/7*a^3*x^(7/2)

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maple [A]  time = 0.00, size = 38, normalized size = 0.75 \[ \frac {2 \left (385 b^{3} x^{6}+1463 a \,b^{2} x^{4}+1995 a^{2} b \,x^{2}+1045 a^{3}\right ) x^{\frac {7}{2}}}{7315} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x^2+a)^3,x)

[Out]

2/7315*x^(7/2)*(385*b^3*x^6+1463*a*b^2*x^4+1995*a^2*b*x^2+1045*a^3)

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maxima [A]  time = 1.34, size = 35, normalized size = 0.69 \[ \frac {2}{19} \, b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, a b^{2} x^{\frac {15}{2}} + \frac {6}{11} \, a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

2/19*b^3*x^(19/2) + 2/5*a*b^2*x^(15/2) + 6/11*a^2*b*x^(11/2) + 2/7*a^3*x^(7/2)

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mupad [B]  time = 0.04, size = 35, normalized size = 0.69 \[ \frac {2\,a^3\,x^{7/2}}{7}+\frac {2\,b^3\,x^{19/2}}{19}+\frac {6\,a^2\,b\,x^{11/2}}{11}+\frac {2\,a\,b^2\,x^{15/2}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a + b*x^2)^3,x)

[Out]

(2*a^3*x^(7/2))/7 + (2*b^3*x^(19/2))/19 + (6*a^2*b*x^(11/2))/11 + (2*a*b^2*x^(15/2))/5

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sympy [A]  time = 10.62, size = 49, normalized size = 0.96 \[ \frac {2 a^{3} x^{\frac {7}{2}}}{7} + \frac {6 a^{2} b x^{\frac {11}{2}}}{11} + \frac {2 a b^{2} x^{\frac {15}{2}}}{5} + \frac {2 b^{3} x^{\frac {19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x**2+a)**3,x)

[Out]

2*a**3*x**(7/2)/7 + 6*a**2*b*x**(11/2)/11 + 2*a*b**2*x**(15/2)/5 + 2*b**3*x**(19/2)/19

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