\(\int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx\) [977]

Optimal result

Integrand size = 22, antiderivative size = 94 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx=\frac {a^2 (d x)^{2+m} \sqrt {c x^2}}{d^2 (2+m) x}+\frac {2 a b (d x)^{3+m} \sqrt {c x^2}}{d^3 (3+m) x}+\frac {b^2 (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 16, 45} \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx=\frac {a^2 \sqrt {c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac {2 a b \sqrt {c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac {b^2 \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]

[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 45

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*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.77 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx=\frac {x (d x)^m \sqrt {c x^2} \left (a^2 \left (12+7 m+m^2\right )+2 a b \left (8+6 m+m^2\right ) x+b^2 \left (6+5 m+m^2\right ) x^2\right )}{(2+m) (3+m) (4+m)} \]

[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] (verified)

Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01

method	result	size
gosper	\(\frac {x \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +5 m \,x^{2} b^{2}+a^{2} m^{2}+12 a b m x +6 b^{2} x^{2}+7 a^{2} m +16 a b x +12 a^{2}\right ) \left (d x \right )^{m} \sqrt {c \,x^{2}}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right )}\)	\(95\)
risch	\(\frac {x \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +5 m \,x^{2} b^{2}+a^{2} m^{2}+12 a b m x +6 b^{2} x^{2}+7 a^{2} m +16 a b x +12 a^{2}\right ) \left (d x \right )^{m} \sqrt {c \,x^{2}}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right )}\)	\(95\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx=\frac {{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 6 \, b^{2}\right )} x^{3} + 2 \, {\left (a b m^{2} + 6 \, a b m + 8 \, a b\right )} x^{2} + {\left (a^{2} m^{2} + 7 \, a^{2} m + 12 \, a^{2}\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (82) = 164\).

Time = 3.08 (sec) , antiderivative size = 483, normalized size of antiderivative = 5.14 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx=\begin {cases} \frac {- \frac {a^{2} \sqrt {c x^{2}}}{2 x^{3}} - \frac {2 a b \sqrt {c x^{2}}}{x^{2}} + \frac {b^{2} \sqrt {c x^{2}} \log {\left (x \right )}}{x}}{d^{4}} & \text {for}\: m = -4 \\\frac {- \frac {a^{2} \sqrt {c x^{2}}}{x^{2}} + \frac {2 a b \sqrt {c x^{2}} \log {\left (x \right )}}{x} + b^{2} \sqrt {c x^{2}}}{d^{3}} & \text {for}\: m = -3 \\\frac {\frac {a^{2} \sqrt {c x^{2}} \log {\left (x \right )}}{x} + 2 a b \sqrt {c x^{2}} + b^{2} \left (\begin {cases} \frac {x \sqrt {c x^{2}}}{2} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{d^{2}} & \text {for}\: m = -2 \\\frac {a^{2} m^{2} x \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {7 a^{2} m x \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {12 a^{2} x \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {2 a b m^{2} x^{2} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {12 a b m x^{2} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {16 a b x^{2} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {b^{2} m^{2} x^{3} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {5 b^{2} m x^{3} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {6 b^{2} x^{3} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 m^{2} + 26 m + 24} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((-a**2*sqrt(c*x**2)/(2*x**3) - 2*a*b*sqrt(c*x**2)/x**2 + b**2*sqrt(c*x**2)*log(x)/x)/d**4, Eq(m, -4
)), ((-a**2*sqrt(c*x**2)/x**2 + 2*a*b*sqrt(c*x**2)*log(x)/x + b**2*sqrt(c*x**2))/d**3, Eq(m, -3)), ((a**2*sqrt
(c*x**2)*log(x)/x + 2*a*b*sqrt(c*x**2) + b**2*Piecewise((x*sqrt(c*x**2)/2, Ne(c, 0)), (0, True)))/d**2, Eq(m,
-2)), (a**2*m**2*x*sqrt(c*x**2)*(d*x)**m/(m**3 + 9*m**2 + 26*m + 24) + 7*a**2*m*x*sqrt(c*x**2)*(d*x)**m/(m**3
+ 9*m**2 + 26*m + 24) + 12*a**2*x*sqrt(c*x**2)*(d*x)**m/(m**3 + 9*m**2 + 26*m + 24) + 2*a*b*m**2*x**2*sqrt(c*x
**2)*(d*x)**m/(m**3 + 9*m**2 + 26*m + 24) + 12*a*b*m*x**2*sqrt(c*x**2)*(d*x)**m/(m**3 + 9*m**2 + 26*m + 24) +
16*a*b*x**2*sqrt(c*x**2)*(d*x)**m/(m**3 + 9*m**2 + 26*m + 24) + b**2*m**2*x**3*sqrt(c*x**2)*(d*x)**m/(m**3 + 9
*m**2 + 26*m + 24) + 5*b**2*m*x**3*sqrt(c*x**2)*(d*x)**m/(m**3 + 9*m**2 + 26*m + 24) + 6*b**2*x**3*sqrt(c*x**2
)*(d*x)**m/(m**3 + 9*m**2 + 26*m + 24), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.68 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx=\frac {b^{2} \sqrt {c} d^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, a b \sqrt {c} d^{m} x^{3} x^{m}}{m + 3} + \frac {a^{2} \sqrt {c} d^{m} x^{2} x^{m}}{m + 2} \]

[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)

Giac [F(-2)]

Exception generated. \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx={\left (d\,x\right )}^m\,\left (\frac {a^2\,x\,\sqrt {c\,x^2}\,\left (m^2+7\,m+12\right )}{m^3+9\,m^2+26\,m+24}+\frac {b^2\,x^3\,\sqrt {c\,x^2}\,\left (m^2+5\,m+6\right )}{m^3+9\,m^2+26\,m+24}+\frac {2\,a\,b\,x^2\,\sqrt {c\,x^2}\,\left (m^2+6\,m+8\right )}{m^3+9\,m^2+26\,m+24}\right ) \]

[In]

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

[Out]

(d*x)^m*((a^2*x*(c*x^2)^(1/2)*(7*m + m^2 + 12))/(26*m + 9*m^2 + m^3 + 24) + (b^2*x^3*(c*x^2)^(1/2)*(5*m + m^2
+ 6))/(26*m + 9*m^2 + m^3 + 24) + (2*a*b*x^2*(c*x^2)^(1/2)*(6*m + m^2 + 8))/(26*m + 9*m^2 + m^3 + 24))