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

Optimal. Leaf size=96 \[ \frac{a^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac{2 a \sqrt{c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac{\sqrt{c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]

[Out]

(a^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b^3*(1 + n)*x) - (2*a*Sqrt[c*x^2]*(a + b*x)^(2 + n))/(b^3*(2 + n)*x) + (S
qrt[c*x^2]*(a + b*x)^(3 + n))/(b^3*(3 + n)*x)

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Rubi [A]  time = 0.0287581, antiderivative size = 96, normalized size of antiderivative = 1., 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} \[ \frac{a^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac{2 a \sqrt{c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac{\sqrt{c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[c*x^2]*(a + b*x)^n,x]

[Out]

(a^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b^3*(1 + n)*x) - (2*a*Sqrt[c*x^2]*(a + b*x)^(2 + n))/(b^3*(2 + n)*x) + (S
qrt[c*x^2]*(a + b*x)^(3 + n))/(b^3*(3 + 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 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 x \sqrt{c x^2} (a+b x)^n \, dx &=\frac{\sqrt{c x^2} \int x^2 (a+b x)^n \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{a^2 (a+b x)^n}{b^2}-\frac{2 a (a+b x)^{1+n}}{b^2}+\frac{(a+b x)^{2+n}}{b^2}\right ) \, dx}{x}\\ &=\frac{a^2 \sqrt{c x^2} (a+b x)^{1+n}}{b^3 (1+n) x}-\frac{2 a \sqrt{c x^2} (a+b x)^{2+n}}{b^3 (2+n) x}+\frac{\sqrt{c x^2} (a+b x)^{3+n}}{b^3 (3+n) x}\\ \end{align*}

Mathematica [A]  time = 0.0460786, size = 68, normalized size = 0.71 \[ \frac{c x (a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 (n+1) (n+2) (n+3) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(a + b*x)^(1 + n)*(2*a^2 - 2*a*b*(1 + n)*x + b^2*(2 + 3*n + n^2)*x^2))/(b^3*(1 + n)*(2 + n)*(3 + n)*Sqrt[
c*x^2])

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Maple [A]  time = 0.004, size = 83, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}{n}^{2}{x}^{2}+3\,{b}^{2}n{x}^{2}-2\,abnx+2\,{b}^{2}{x}^{2}-2\,abx+2\,{a}^{2} \right ) }{x{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }\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]

(b*x+a)^(1+n)*(b^2*n^2*x^2+3*b^2*n*x^2-2*a*b*n*x+2*b^2*x^2-2*a*b*x+2*a^2)*(c*x^2)^(1/2)/x/b^3/(n^3+6*n^2+11*n+
6)

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Maxima [A]  time = 1.02689, size = 108, normalized size = 1.12 \begin{align*} \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} \sqrt{c} x^{3} +{\left (n^{2} + n\right )} a b^{2} \sqrt{c} x^{2} - 2 \, a^{2} b \sqrt{c} n x + 2 \, a^{3} \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \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]

((n^2 + 3*n + 2)*b^3*sqrt(c)*x^3 + (n^2 + n)*a*b^2*sqrt(c)*x^2 - 2*a^2*b*sqrt(c)*n*x + 2*a^3*sqrt(c))*(b*x + a
)^n/((n^3 + 6*n^2 + 11*n + 6)*b^3)

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Fricas [A]  time = 1.44238, size = 209, normalized size = 2.18 \begin{align*} -\frac{{\left (2 \, a^{2} b n x -{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} - 2 \, a^{3} -{\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}\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]

-(2*a^2*b*n*x - (b^3*n^2 + 3*b^3*n + 2*b^3)*x^3 - 2*a^3 - (a*b^2*n^2 + a*b^2*n)*x^2)*sqrt(c*x^2)*(b*x + a)^n/(
(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3)*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 [B]  time = 1.08288, size = 270, normalized size = 2.81 \begin{align*} -{\left (\frac{2 \, a^{3} a^{n} \mathrm{sgn}\left (x\right )}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} - \frac{{\left (b x + a\right )}^{n} b^{3} n^{2} x^{3} \mathrm{sgn}\left (x\right ) +{\left (b x + a\right )}^{n} a b^{2} n^{2} x^{2} \mathrm{sgn}\left (x\right ) + 3 \,{\left (b x + a\right )}^{n} b^{3} n x^{3} \mathrm{sgn}\left (x\right ) +{\left (b x + a\right )}^{n} a b^{2} n x^{2} \mathrm{sgn}\left (x\right ) + 2 \,{\left (b x + a\right )}^{n} b^{3} x^{3} \mathrm{sgn}\left (x\right ) - 2 \,{\left (b x + a\right )}^{n} a^{2} b n x \mathrm{sgn}\left (x\right ) + 2 \,{\left (b x + a\right )}^{n} a^{3} \mathrm{sgn}\left (x\right )}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}}\right )} \sqrt{c} \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]

-(2*a^3*a^n*sgn(x)/(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3) - ((b*x + a)^n*b^3*n^2*x^3*sgn(x) + (b*x + a)^n*a*
b^2*n^2*x^2*sgn(x) + 3*(b*x + a)^n*b^3*n*x^3*sgn(x) + (b*x + a)^n*a*b^2*n*x^2*sgn(x) + 2*(b*x + a)^n*b^3*x^3*s
gn(x) - 2*(b*x + a)^n*a^2*b*n*x*sgn(x) + 2*(b*x + a)^n*a^3*sgn(x))/(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3))*s
qrt(c)