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

Optimal. Leaf size=63 \[ \frac {\sqrt {c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac {a \sqrt {c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]

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Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 43} \begin {gather*} \frac {\sqrt {c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac {a \sqrt {c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]*(a + b*x)^n,x]

[Out]

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

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Mathematica [A]  time = 0.03, size = 44, normalized size = 0.70 \begin {gather*} \frac {c x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*x^2]*(a + b*x)^n,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c x^2} (a+b x)^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 63, normalized size = 1.00 \begin {gather*} \frac {{\left (a b n x + {\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.14, size = 119, normalized size = 1.89 \begin {gather*} {\left (\frac {a^{2} a^{n} \mathrm {sgn}\relax (x)}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} + \frac {{\left (b x + a\right )}^{n} b^{2} n x^{2} \mathrm {sgn}\relax (x) + {\left (b x + a\right )}^{n} a b n x \mathrm {sgn}\relax (x) + {\left (b x + a\right )}^{n} b^{2} x^{2} \mathrm {sgn}\relax (x) - {\left (b x + a\right )}^{n} a^{2} \mathrm {sgn}\relax (x)}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}}\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 46, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {c \,x^{2}}\, \left (-x n b -b x +a \right ) \left (b x +a \right )^{n +1}}{\left (n^{2}+3 n +2\right ) b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.45, size = 51, normalized size = 0.81 \begin {gather*} \frac {{\left (b^{2} \sqrt {c} {\left (n + 1\right )} x^{2} + a b \sqrt {c} n x - a^{2} \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.22, size = 85, normalized size = 1.35 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{n^2+3\,n+2}-\frac {a^2\,\sqrt {c\,x^2}}{b^2\,\left (n^2+3\,n+2\right )}+\frac {a\,n\,x\,\sqrt {c\,x^2}}{b\,\left (n^2+3\,n+2\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(1/2)*(a + b*x)^n,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {a^{n} \sqrt {c} x \sqrt {x^{2}}}{2} & \text {for}\: b = 0 \\\int \frac {\sqrt {c x^{2}}}{\left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {\sqrt {c x^{2}}}{a + b x}\, dx & \text {for}\: n = -1 \\- \frac {a^{2} \sqrt {c} \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} + \frac {a b \sqrt {c} n x \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} + \frac {b^{2} \sqrt {c} n x^{2} \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} + \frac {b^{2} \sqrt {c} x^{2} \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**n*sqrt(c)*x*sqrt(x**2)/2, Eq(b, 0)), (Integral(sqrt(c*x**2)/(a + b*x)**2, x), Eq(n, -2)), (Integ
ral(sqrt(c*x**2)/(a + b*x), x), Eq(n, -1)), (-a**2*sqrt(c)*(a + b*x)**n*sqrt(x**2)/(b**2*n**2*x + 3*b**2*n*x +
2*b**2*x) + a*b*sqrt(c)*n*x*(a + b*x)**n*sqrt(x**2)/(b**2*n**2*x + 3*b**2*n*x + 2*b**2*x) + b**2*sqrt(c)*n*x*
*2*(a + b*x)**n*sqrt(x**2)/(b**2*n**2*x + 3*b**2*n*x + 2*b**2*x) + b**2*sqrt(c)*x**2*(a + b*x)**n*sqrt(x**2)/(
b**2*n**2*x + 3*b**2*n*x + 2*b**2*x), True))

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