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

Optimal. Leaf size=81 \[ \frac{a^2 x (d x)^m}{m \sqrt{c x^2}}+\frac{2 a b x (d x)^{m+1}}{d (m+1) \sqrt{c x^2}}+\frac{b^2 x (d x)^{m+2}}{d^2 (m+2) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0355124, antiderivative size = 81, 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} \[ \frac{a^2 x (d x)^m}{m \sqrt{c x^2}}+\frac{2 a b x (d x)^{m+1}}{d (m+1) \sqrt{c x^2}}+\frac{b^2 x (d x)^{m+2}}{d^2 (m+2) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.032731, size = 62, normalized size = 0.77 \[ \frac{x (d x)^m \left (a^2 \left (m^2+3 m+2\right )+2 a b m (m+2) x+b^2 m (m+1) x^2\right )}{m (m+1) (m+2) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 79, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{2}{x}^{2}{m}^{2}+2\,abx{m}^{2}+{b}^{2}{x}^{2}m+{a}^{2}{m}^{2}+4\,abxm+3\,{a}^{2}m+2\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 2+m \right ) \left ( 1+m \right ) m}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x+b^2*m*x^2+a^2*m^2+4*a*b*m*x+3*a^2*m+2*a^2)*(d*x)^m/(2+m)/(1+m)/m/(c*x^2)^(1/2)

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Maxima [A]  time = 1.08843, size = 77, normalized size = 0.95 \begin{align*} \frac{b^{2} d^{m} x^{2} x^{m}}{\sqrt{c}{\left (m + 2\right )}} + \frac{2 \, a b d^{m} x x^{m}}{\sqrt{c}{\left (m + 1\right )}} + \frac{a^{2} d^{m} x^{m}}{\sqrt{c} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42444, size = 174, normalized size = 2.15 \begin{align*} \frac{{\left (a^{2} m^{2} + 3 \, a^{2} m +{\left (b^{2} m^{2} + b^{2} m\right )} x^{2} + 2 \, a^{2} + 2 \,{\left (a b m^{2} + 2 \, a b m\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{{\left (c m^{3} + 3 \, c m^{2} + 2 \, c m\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*m^2 + 3*a^2*m + (b^2*m^2 + b^2*m)*x^2 + 2*a^2 + 2*(a*b*m^2 + 2*a*b*m)*x)*sqrt(c*x^2)*(d*x)^m/((c*m^3 + 3*
c*m^2 + 2*c*m)*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((d*x)**m*(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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