3.3.72 \(\int x^{7/2} (a+b x^2)^2 \, dx\) [272]

Optimal. Leaf size=36 \[ \frac {2}{9} a^2 x^{9/2}+\frac {4}{13} a b x^{13/2}+\frac {2}{17} b^2 x^{17/2} \]

[Out]

2/9*a^2*x^(9/2)+4/13*a*b*x^(13/2)+2/17*b^2*x^(17/2)

________________________________________________________________________________________

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}{9} a^2 x^{9/2}+\frac {4}{13} a b x^{13/2}+\frac {2}{17} b^2 x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(a + b*x^2)^2,x]

[Out]

(2*a^2*x^(9/2))/9 + (4*a*b*x^(13/2))/13 + (2*b^2*x^(17/2))/17

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^{7/2} \left (a+b x^2\right )^2 \, dx &=\int \left (a^2 x^{7/2}+2 a b x^{11/2}+b^2 x^{15/2}\right ) \, dx\\ &=\frac {2}{9} a^2 x^{9/2}+\frac {4}{13} a b x^{13/2}+\frac {2}{17} b^2 x^{17/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 30, normalized size = 0.83 \begin {gather*} \frac {2 x^{9/2} \left (221 a^2+306 a b x^2+117 b^2 x^4\right )}{1989} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(a + b*x^2)^2,x]

[Out]

(2*x^(9/2)*(221*a^2 + 306*a*b*x^2 + 117*b^2*x^4))/1989

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 25, normalized size = 0.69

method	result	size
derivativedivides	\(\frac {2 a^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,x^{\frac {13}{2}}}{13}+\frac {2 b^{2} x^{\frac {17}{2}}}{17}\)	\(25\)
default	\(\frac {2 a^{2} x^{\frac {9}{2}}}{9}+\frac {4 a b \,x^{\frac {13}{2}}}{13}+\frac {2 b^{2} x^{\frac {17}{2}}}{17}\)	\(25\)
gosper	\(\frac {2 x^{\frac {9}{2}} \left (117 b^{2} x^{4}+306 a b \,x^{2}+221 a^{2}\right )}{1989}\)	\(27\)
trager	\(\frac {2 x^{\frac {9}{2}} \left (117 b^{2} x^{4}+306 a b \,x^{2}+221 a^{2}\right )}{1989}\)	\(27\)
risch	\(\frac {2 x^{\frac {9}{2}} \left (117 b^{2} x^{4}+306 a b \,x^{2}+221 a^{2}\right )}{1989}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/9*a^2*x^(9/2)+4/13*a*b*x^(13/2)+2/17*b^2*x^(17/2)

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{17} \, b^{2} x^{\frac {17}{2}} + \frac {4}{13} \, a b x^{\frac {13}{2}} + \frac {2}{9} \, a^{2} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2,x, algorithm="maxima")

[Out]

2/17*b^2*x^(17/2) + 4/13*a*b*x^(13/2) + 2/9*a^2*x^(9/2)

________________________________________________________________________________________

Fricas [A]
time = 1.59, size = 29, normalized size = 0.81 \begin {gather*} \frac {2}{1989} \, {\left (117 \, b^{2} x^{8} + 306 \, a b x^{6} + 221 \, a^{2} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2,x, algorithm="fricas")

[Out]

2/1989*(117*b^2*x^8 + 306*a*b*x^6 + 221*a^2*x^4)*sqrt(x)

________________________________________________________________________________________

Sympy [A]
time = 0.70, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 a^{2} x^{\frac {9}{2}}}{9} + \frac {4 a b x^{\frac {13}{2}}}{13} + \frac {2 b^{2} x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(b*x**2+a)**2,x)

[Out]

2*a**2*x**(9/2)/9 + 4*a*b*x**(13/2)/13 + 2*b**2*x**(17/2)/17

________________________________________________________________________________________

Giac [A]
time = 0.96, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{17} \, b^{2} x^{\frac {17}{2}} + \frac {4}{13} \, a b x^{\frac {13}{2}} + \frac {2}{9} \, a^{2} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(b*x^2+a)^2,x, algorithm="giac")

[Out]

2/17*b^2*x^(17/2) + 4/13*a*b*x^(13/2) + 2/9*a^2*x^(9/2)

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 25, normalized size = 0.69 \begin {gather*} x^{9/2}\,\left (\frac {2\,a^2}{9}+\frac {4\,a\,b\,x^2}{13}+\frac {2\,b^2\,x^4}{17}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(a + b*x^2)^2,x)

[Out]

x^(9/2)*((2*a^2)/9 + (2*b^2*x^4)/17 + (4*a*b*x^2)/13)

________________________________________________________________________________________