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

Optimal. Leaf size=30 \[ \frac{\sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0056715, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{\sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^n)/x,x]

[Out]

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

Mathematica [A]  time = 0.0125384, size = 29, normalized size = 0.97 \[ \frac{c x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 29, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ) x}\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05858, size = 38, normalized size = 1.27 \begin{align*} \frac{{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.33024, size = 66, normalized size = 2.2 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [A]  time = 1.06366, size = 57, normalized size = 1.9 \begin{align*} -\sqrt{c}{\left (\frac{a^{n + 1} \mathrm{sgn}\left (x\right )}{b n + b} - \frac{{\left (b x + a\right )}^{n + 1} \mathrm{sgn}\left (x\right )}{b{\left (n + 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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