3.717 \(\int x^{5/2} (a+c x^4) \, dx\)

Optimal. Leaf size=21 \[ \frac {2}{7} a x^{7/2}+\frac {2}{15} c x^{15/2} \]

[Out]

2/7*a*x^(7/2)+2/15*c*x^(15/2)

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \frac {2}{7} a x^{7/2}+\frac {2}{15} c x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + c*x^4),x]

[Out]

(2*a*x^(7/2))/7 + (2*c*x^(15/2))/15

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^{5/2} \left (a+c x^4\right ) \, dx &=\int \left (a x^{5/2}+c x^{13/2}\right ) \, dx\\ &=\frac {2}{7} a x^{7/2}+\frac {2}{15} c x^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ \frac {2}{7} a x^{7/2}+\frac {2}{15} c x^{15/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + c*x^4),x]

[Out]

(2*a*x^(7/2))/7 + (2*c*x^(15/2))/15

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fricas [A]  time = 0.48, size = 18, normalized size = 0.86 \[ \frac {2}{105} \, {\left (7 \, c x^{7} + 15 \, a x^{3}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+a),x, algorithm="fricas")

[Out]

2/105*(7*c*x^7 + 15*a*x^3)*sqrt(x)

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giac [A]  time = 0.17, size = 13, normalized size = 0.62 \[ \frac {2}{15} \, c x^{\frac {15}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+a),x, algorithm="giac")

[Out]

2/15*c*x^(15/2) + 2/7*a*x^(7/2)

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maple [A]  time = 0.00, size = 16, normalized size = 0.76 \[ \frac {2 \left (7 c \,x^{4}+15 a \right ) x^{\frac {7}{2}}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(c*x^4+a),x)

[Out]

2/105*x^(7/2)*(7*c*x^4+15*a)

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maxima [A]  time = 1.36, size = 13, normalized size = 0.62 \[ \frac {2}{15} \, c x^{\frac {15}{2}} + \frac {2}{7} \, a x^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+a),x, algorithm="maxima")

[Out]

2/15*c*x^(15/2) + 2/7*a*x^(7/2)

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mupad [B]  time = 1.05, size = 15, normalized size = 0.71 \[ \frac {2\,x^{7/2}\,\left (7\,c\,x^4+15\,a\right )}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a + c*x^4),x)

[Out]

(2*x^(7/2)*(15*a + 7*c*x^4))/105

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sympy [A]  time = 7.94, size = 19, normalized size = 0.90 \[ \frac {2 a x^{\frac {7}{2}}}{7} + \frac {2 c x^{\frac {15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(c*x**4+a),x)

[Out]

2*a*x**(7/2)/7 + 2*c*x**(15/2)/15

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