3.8.66 \(\int \frac {\sqrt {c x^2} (a+b x)^2}{x} \, dx\)

Optimal. Leaf size=26 \[ \frac {\sqrt {c x^2} (a+b x)^3}{3 b x} \]

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \begin {gather*} \frac {\sqrt {c x^2} (a+b x)^3}{3 b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x,x]

[Out]

(Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.96 \begin {gather*} \frac {c x (a+b x)^3}{3 b \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(a + b*x)^3)/(3*b*Sqrt[c*x^2])

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IntegrateAlgebraic [A]  time = 0.03, size = 31, normalized size = 1.19 \begin {gather*} \frac {1}{3} \sqrt {c x^2} \left (3 a^2+3 a b x+b^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.08, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, {\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.97, size = 29, normalized size = 1.12 \begin {gather*} \frac {1}{3} \, {\left (\frac {{\left (b x + a\right )}^{3} \mathrm {sgn}\relax (x)}{b} - \frac {a^{3} \mathrm {sgn}\relax (x)}{b}\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x,x, algorithm="giac")

[Out]

1/3*((b*x + a)^3*sgn(x)/b - a^3*sgn(x)/b)*sqrt(c)

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maple [A]  time = 0.00, size = 28, normalized size = 1.08 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) \sqrt {c \,x^{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(c*x^2)^(1/2)/x,x)

[Out]

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

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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((b*x+a)^2*(c*x^2)^(1/2)/x,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 [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(1/2)*(a + b*x)^2)/x,x)

[Out]

int(((c*x^2)^(1/2)*(a + b*x)^2)/x, x)

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sympy [B]  time = 0.30, size = 51, normalized size = 1.96 \begin {gather*} a^{2} \sqrt {c} \sqrt {x^{2}} + a b \sqrt {c} x \sqrt {x^{2}} + \frac {b^{2} \sqrt {c} x^{2} \sqrt {x^{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(c*x**2)**(1/2)/x,x)

[Out]

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

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