3.2.74 \(\int x^{7/2} (b x^2+c x^4) \, dx\)

Optimal. Leaf size=21 \[ \frac {2}{13} b x^{13/2}+\frac {2}{17} c x^{17/2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \begin {gather*} \frac {2}{13} b x^{13/2}+\frac {2}{17} c x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(13/2))/13 + (2*c*x^(17/2))/17

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{13} b x^{13/2}+\frac {2}{17} c x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(b*x^2 + c*x^4),x]

[Out]

(2*b*x^(13/2))/13 + (2*c*x^(17/2))/17

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{221} \left (17 b x^{13/2}+13 c x^{17/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(7/2)*(b*x^2 + c*x^4),x]

[Out]

(2*(17*b*x^(13/2) + 13*c*x^(17/2)))/221

________________________________________________________________________________________

fricas [A]  time = 1.56, size = 18, normalized size = 0.86 \begin {gather*} \frac {2}{221} \, {\left (13 \, c x^{8} + 17 \, b x^{6}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

2/221*(13*c*x^8 + 17*b*x^6)*sqrt(x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{17} \, c x^{\frac {17}{2}} + \frac {2}{13} \, b x^{\frac {13}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2),x, algorithm="giac")

[Out]

2/17*c*x^(17/2) + 2/13*b*x^(13/2)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 16, normalized size = 0.76 \begin {gather*} \frac {2 \left (13 c \,x^{2}+17 b \right ) x^{\frac {13}{2}}}{221} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(c*x^4+b*x^2),x)

[Out]

2/221*x^(13/2)*(13*c*x^2+17*b)

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{17} \, c x^{\frac {17}{2}} + \frac {2}{13} \, b x^{\frac {13}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

2/17*c*x^(17/2) + 2/13*b*x^(13/2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 15, normalized size = 0.71 \begin {gather*} \frac {2\,x^{13/2}\,\left (13\,c\,x^2+17\,b\right )}{221} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(b*x^2 + c*x^4),x)

[Out]

(2*x^(13/2)*(17*b + 13*c*x^2))/221

________________________________________________________________________________________

sympy [A]  time = 11.34, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 b x^{\frac {13}{2}}}{13} + \frac {2 c x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(c*x**4+b*x**2),x)

[Out]

2*b*x**(13/2)/13 + 2*c*x**(17/2)/17

________________________________________________________________________________________