3.8.78 \(\int \frac {(c x^2)^{5/2} (a+b x)}{x^3} \, dx\) [778]

Optimal. Leaf size=41 \[ \frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2} \]

[Out]

1/3*a*c^2*x^2*(c*x^2)^(1/2)+1/4*b*c^2*x^3*(c*x^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \begin {gather*} \frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(5/2)*(a + b*x))/x^3,x]

[Out]

(a*c^2*x^2*Sqrt[c*x^2])/3 + (b*c^2*x^3*Sqrt[c*x^2])/4

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^2 (a+b x) \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a x^2+b x^3\right ) \, dx}{x}\\ &=\frac {1}{3} a c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b c^2 x^3 \sqrt {c x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 27, normalized size = 0.66 \begin {gather*} \frac {1}{12} c^2 x^2 \sqrt {c x^2} (4 a+3 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c*x^2)^(5/2)*(a + b*x))/x^3,x]

[Out]

(c^2*x^2*Sqrt[c*x^2]*(4*a + 3*b*x))/12

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Maple [A]
time = 0.02, size = 21, normalized size = 0.51

method	result	size
gosper	\(\frac {\left (3 b x +4 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{2}}\)	\(21\)
default	\(\frac {\left (3 b x +4 a \right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{2}}\)	\(21\)
risch	\(\frac {a \,c^{2} x^{2} \sqrt {c \,x^{2}}}{3}+\frac {b \,c^{2} x^{3} \sqrt {c \,x^{2}}}{4}\)	\(34\)
trager	\(\frac {c^{2} \left (3 b \,x^{3}+4 a \,x^{2}+3 x^{2} b +4 a x +3 b x +4 a +3 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 x}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)*(b*x+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

1/12/x^2*(3*b*x+4*a)*(c*x^2)^(5/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 0.87, size = 28, normalized size = 0.68 \begin {gather*} \frac {1}{12} \, {\left (3 \, b c^{2} x^{3} + 4 \, a c^{2} x^{2}\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b*c^2*x^3 + 4*a*c^2*x^2)*sqrt(c*x^2)

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Sympy [A]
time = 0.33, size = 27, normalized size = 0.66 \begin {gather*} \frac {a \left (c x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} + \frac {b \left (c x^{2}\right )^{\frac {5}{2}}}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(c*x**2)**(5/2)/(3*x**2) + b*(c*x**2)**(5/2)/(4*x)

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Giac [A]
time = 1.76, size = 28, normalized size = 0.68 \begin {gather*} \frac {1}{12} \, {\left (3 \, b c^{2} x^{4} \mathrm {sgn}\left (x\right ) + 4 \, a c^{2} x^{3} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b*c^2*x^4*sgn(x) + 4*a*c^2*x^3*sgn(x))*sqrt(c)

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Mupad [B]
time = 0.27, size = 25, normalized size = 0.61 \begin {gather*} \frac {c^{5/2}\,\left (4\,a\,\sqrt {x^6}+3\,b\,x^3\,\sqrt {x^2}\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(5/2)*(a + b*x))/x^3,x)

[Out]

(c^(5/2)*(4*a*(x^6)^(1/2) + 3*b*x^3*(x^2)^(1/2)))/12

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