3.8.27 \(\int \frac {(c x^2)^{3/2} (a+b x)}{x^2} \, dx\)

Optimal. Leaf size=35 \[ \frac {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, 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, 43} \begin {gather*} \frac {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*x^2)^(3/2)*(a + b*x))/x^2,x]

[Out]

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

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 43

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

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Mathematica [A]  time = 0.00, size = 23, normalized size = 0.66 \begin {gather*} \frac {1}{6} c x \sqrt {c x^2} (3 a+2 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*Sqrt[c*x^2]*(3*a + 2*b*x))/6

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IntegrateAlgebraic [A]  time = 0.02, size = 24, normalized size = 0.69 \begin {gather*} \frac {\left (c x^2\right )^{3/2} (3 a+2 b x)}{6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*x^2)^(3/2)*(3*a + 2*b*x))/(6*x)

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fricas [A]  time = 1.01, size = 22, normalized size = 0.63 \begin {gather*} \frac {1}{6} \, {\left (2 \, b c x^{2} + 3 \, a c x\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*c*x^2 + 3*a*c*x)*sqrt(c*x^2)

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giac [A]  time = 1.08, size = 22, normalized size = 0.63 \begin {gather*} \frac {1}{6} \, {\left (2 \, b x^{3} \mathrm {sgn}\relax (x) + 3 \, a x^{2} \mathrm {sgn}\relax (x)\right )} c^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*x^3*sgn(x) + 3*a*x^2*sgn(x))*c^(3/2)

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maple [A]  time = 0.00, size = 21, normalized size = 0.60 \begin {gather*} \frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{6 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^(3/2)*(b*x+a)/x^2,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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mupad [B]  time = 0.27, size = 20, normalized size = 0.57 \begin {gather*} \frac {c^{3/2}\,\left (2\,b\,\sqrt {x^6}+3\,a\,x\,\relax |x|\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(3/2)*(a + b*x))/x^2,x)

[Out]

(c^(3/2)*(2*b*(x^6)^(1/2) + 3*a*x*abs(x)))/6

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sympy [A]  time = 0.57, size = 31, normalized size = 0.89 \begin {gather*} \frac {a c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{2 x} + \frac {b c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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