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

Optimal. Leaf size=94 \[ \frac{a^2 \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{2 a b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac{b^2 \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]

[Out]

(a^2*(d*x)^(2 + m)*Sqrt[c*x^2])/(d^2*(2 + m)*x) + (2*a*b*(d*x)^(3 + m)*Sqrt[c*x^2])/(d^3*(3 + m)*x) + (b^2*(d*
x)^(4 + m)*Sqrt[c*x^2])/(d^4*(4 + m)*x)

________________________________________________________________________________________

Rubi [A]  time = 0.0410875, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {15, 16, 43} \[ \frac{a^2 \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{2 a b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac{b^2 \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(d*x)^(2 + m)*Sqrt[c*x^2])/(d^2*(2 + m)*x) + (2*a*b*(d*x)^(3 + m)*Sqrt[c*x^2])/(d^3*(3 + m)*x) + (b^2*(d*
x)^(4 + m)*Sqrt[c*x^2])/(d^4*(4 + m)*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 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 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 (d x)^m \sqrt{c x^2} (a+b x)^2 \, dx &=\frac{\sqrt{c x^2} \int x (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int (d x)^{1+m} (a+b x)^2 \, dx}{d x}\\ &=\frac{\sqrt{c x^2} \int \left (a^2 (d x)^{1+m}+\frac{2 a b (d x)^{2+m}}{d}+\frac{b^2 (d x)^{3+m}}{d^2}\right ) \, dx}{d x}\\ &=\frac{a^2 (d x)^{2+m} \sqrt{c x^2}}{d^2 (2+m) x}+\frac{2 a b (d x)^{3+m} \sqrt{c x^2}}{d^3 (3+m) x}+\frac{b^2 (d x)^{4+m} \sqrt{c x^2}}{d^4 (4+m) x}\\ \end{align*}

Mathematica [A]  time = 0.049885, size = 72, normalized size = 0.77 \[ \frac{x \sqrt{c x^2} (d x)^m \left (a^2 \left (m^2+7 m+12\right )+2 a b \left (m^2+6 m+8\right ) x+b^2 \left (m^2+5 m+6\right ) x^2\right )}{(m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(d*x)^m*Sqrt[c*x^2]*(a^2*(12 + 7*m + m^2) + 2*a*b*(8 + 6*m + m^2)*x + b^2*(6 + 5*m + m^2)*x^2))/((2 + m)*(3
+ m)*(4 + m))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 95, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+5\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+12\,abmx+6\,{b}^{2}{x}^{2}+7\,{a}^{2}m+16\,abx+12\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) }\sqrt{c{x}^{2}}} \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)^2,x)

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x+5*b^2*m*x^2+a^2*m^2+12*a*b*m*x+6*b^2*x^2+7*a^2*m+16*a*b*x+12*a^2)*(d*x)^m*(c*x^2)^(
1/2)/(4+m)/(3+m)/(2+m)

________________________________________________________________________________________

Maxima [A]  time = 1.08932, size = 86, normalized size = 0.91 \begin{align*} \frac{b^{2} \sqrt{c} d^{m} x^{4} x^{m}}{m + 4} + \frac{2 \, a b \sqrt{c} d^{m} x^{3} x^{m}}{m + 3} + \frac{a^{2} \sqrt{c} d^{m} x^{2} x^{m}}{m + 2} \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)^2,x, algorithm="maxima")

[Out]

b^2*sqrt(c)*d^m*x^4*x^m/(m + 4) + 2*a*b*sqrt(c)*d^m*x^3*x^m/(m + 3) + a^2*sqrt(c)*d^m*x^2*x^m/(m + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.35785, size = 203, normalized size = 2.16 \begin{align*} \frac{{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 6 \, b^{2}\right )} x^{3} + 2 \,{\left (a b m^{2} + 6 \, a b m + 8 \, a b\right )} x^{2} +{\left (a^{2} m^{2} + 7 \, a^{2} m + 12 \, a^{2}\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \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)^2,x, algorithm="fricas")

[Out]

((b^2*m^2 + 5*b^2*m + 6*b^2)*x^3 + 2*(a*b*m^2 + 6*a*b*m + 8*a*b)*x^2 + (a^2*m^2 + 7*a^2*m + 12*a^2)*x)*sqrt(c*
x^2)*(d*x)^m/(m^3 + 9*m^2 + 26*m + 24)

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError