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

Optimal. Leaf size=64 \[ \frac{\sqrt{c x^2} (d x)^{m+2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+2,-n;m+3;-\frac{b x}{a}\right )}{d^2 (m+2) x} \]

[Out]

((d*x)^(2 + m)*Sqrt[c*x^2]*(a + b*x)^n*Hypergeometric2F1[2 + m, -n, 3 + m, -((b*x)/a)])/(d^2*(2 + m)*x*(1 + (b
*x)/a)^n)

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Rubi [A]  time = 0.0281022, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 16, 66, 64} \[ \frac{\sqrt{c x^2} (d x)^{m+2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+2,-n;m+3;-\frac{b x}{a}\right )}{d^2 (m+2) x} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*Sqrt[c*x^2]*(a + b*x)^n,x]

[Out]

((d*x)^(2 + m)*Sqrt[c*x^2]*(a + b*x)^n*Hypergeometric2F1[2 + m, -n, 3 + m, -((b*x)/a)])/(d^2*(2 + m)*x*(1 + (b
*x)/a)^n)

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 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c^IntPart[n]*(c + d*x)^FracPart[n])/(1 + (d
*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0]))
||  !RationalQ[n])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int (d x)^m \sqrt{c x^2} (a+b x)^n \, dx &=\frac{\sqrt{c x^2} \int x (d x)^m (a+b x)^n \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int (d x)^{1+m} (a+b x)^n \, dx}{d x}\\ &=\frac{\left (\sqrt{c x^2} (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n}\right ) \int (d x)^{1+m} \left (1+\frac{b x}{a}\right )^n \, dx}{d x}\\ &=\frac{(d x)^{2+m} \sqrt{c x^2} (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n} \, _2F_1\left (2+m,-n;3+m;-\frac{b x}{a}\right )}{d^2 (2+m) x}\\ \end{align*}

Mathematica [A]  time = 0.0140299, size = 57, normalized size = 0.89 \[ \frac{x \sqrt{c x^2} (d x)^m (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+2,-n;m+3;-\frac{b x}{a}\right )}{m+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(d*x)^m*Sqrt[c*x^2]*(a + b*x)^n*Hypergeometric2F1[2 + m, -n, 3 + m, -((b*x)/a)])/((2 + m)*(1 + (b*x)/a)^n)

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{c{x}^{2}} \left ( bx+a \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2)*(b*x + a)^n*(d*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c*x**2)*(d*x)**m*(a + b*x)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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