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

Optimal. Leaf size=28 \[ \frac{x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]

[Out]

(x*(a + b*x)^(1 + n))/(b*(1 + n)*Sqrt[c*x^2])

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Rubi [A]  time = 0.0048816, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 32} \[ \frac{x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^n)/Sqrt[c*x^2],x]

[Out]

(x*(a + b*x)^(1 + n))/(b*(1 + n)*Sqrt[c*x^2])

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

Mathematica [A]  time = 0.0106076, size = 28, normalized size = 1. \[ \frac{x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x)^(1 + n))/(b*(1 + n)*Sqrt[c*x^2])

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Maple [A]  time = 0., size = 27, normalized size = 1. \begin{align*}{\frac{x \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ) }{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01808, size = 42, normalized size = 1.5 \begin{align*} \frac{{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{b c{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6154, size = 72, normalized size = 2.57 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{{\left (b c n + b c\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n} x}{\sqrt{c x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n*x/sqrt(c*x^2), x)