\(\int \frac {(d x)^m (a+b x)^2}{(c x^2)^{3/2}} \, dx\) [979]

Optimal result

Integrand size = 22, antiderivative size = 93 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=-\frac {a^2 d^2 x (d x)^{-2+m}}{c (2-m) \sqrt {c x^2}}-\frac {2 a b d x (d x)^{-1+m}}{c (1-m) \sqrt {c x^2}}+\frac {b^2 x (d x)^m}{c m \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 93, 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} \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=-\frac {a^2 d^2 x (d x)^{m-2}}{c (2-m) \sqrt {c x^2}}-\frac {2 a b d x (d x)^{m-1}}{c (1-m) \sqrt {c x^2}}+\frac {b^2 x (d x)^m}{c m \sqrt {c x^2}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {x (d x)^m \left (a^2 (-1+m) m+2 a b (-2+m) m x+b^2 \left (2-3 m+m^2\right ) x^2\right )}{(-2+m) (-1+m) m \left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.89

method	result	size
gosper	\(\frac {x \left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x -3 m \,x^{2} b^{2}+a^{2} m^{2}-4 a b m x +2 b^{2} x^{2}-a^{2} m \right ) \left (d x \right )^{m}}{m \left (-1+m \right ) \left (-2+m \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(83\)
risch	\(\frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x -3 m \,x^{2} b^{2}+a^{2} m^{2}-4 a b m x +2 b^{2} x^{2}-a^{2} m \right ) \left (d x \right )^{m}}{c x \sqrt {c \,x^{2}}\, m \left (-1+m \right ) \left (-2+m \right )}\)	\(88\)

[In]

int((d*x)^m*(b*x+a)^2/(c*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {{\left (a^{2} m^{2} - a^{2} m + {\left (b^{2} m^{2} - 3 \, b^{2} m + 2 \, b^{2}\right )} x^{2} + 2 \, {\left (a b m^{2} - 2 \, a b m\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{{\left (c^{2} m^{3} - 3 \, c^{2} m^{2} + 2 \, c^{2} m\right )} x^{3}} \]

[In]

integrate((d*x)^m*(b*x+a)^2/(c*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (78) = 156\).

Time = 3.97 (sec) , antiderivative size = 532, normalized size of antiderivative = 5.72 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {a^{2} x}{2 \left (c x^{2}\right )^{\frac {3}{2}}} + 2 a b \left (\begin {cases} \tilde {\infty } x^{2} & \text {for}\: c = 0 \\- \frac {1}{c \sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) + \frac {b^{2} x^{3} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: m = 0 \\d \left (a^{2} \left (\begin {cases} \tilde {\infty } x^{2} & \text {for}\: c = 0 \\- \frac {1}{c \sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) + \frac {2 a b x^{3} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}}\right ) & \text {for}\: m = 1 \\d^{2} \left (\frac {a^{2} x^{3} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {2 a b x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{5}}{2 \left (c x^{2}\right )^{\frac {3}{2}}}\right ) & \text {for}\: m = 2 \\\frac {a^{2} m^{2} x \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 2 m \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {a^{2} m x \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 2 m \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {2 a b m^{2} x^{2} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 2 m \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {4 a b m x^{2} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 2 m \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} m^{2} x^{3} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 2 m \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {3 b^{2} m x^{3} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 2 m \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {2 b^{2} x^{3} \left (d x\right )^{m}}{m^{3} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 2 m \left (c x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x)**m*(b*x+a)**2/(c*x**2)**(3/2),x)

[Out]

Piecewise((-a**2*x/(2*(c*x**2)**(3/2)) + 2*a*b*Piecewise((zoo*x**2, Eq(c, 0)), (-1/(c*sqrt(c*x**2)), True)) +
b**2*x**3*log(x)/(c*x**2)**(3/2), Eq(m, 0)), (d*(a**2*Piecewise((zoo*x**2, Eq(c, 0)), (-1/(c*sqrt(c*x**2)), Tr
ue)) + 2*a*b*x**3*log(x)/(c*x**2)**(3/2) + b**2*x**4/(c*x**2)**(3/2)), Eq(m, 1)), (d**2*(a**2*x**3*log(x)/(c*x
**2)**(3/2) + 2*a*b*x**4/(c*x**2)**(3/2) + b**2*x**5/(2*(c*x**2)**(3/2))), Eq(m, 2)), (a**2*m**2*x*(d*x)**m/(m
**3*(c*x**2)**(3/2) - 3*m**2*(c*x**2)**(3/2) + 2*m*(c*x**2)**(3/2)) - a**2*m*x*(d*x)**m/(m**3*(c*x**2)**(3/2)
- 3*m**2*(c*x**2)**(3/2) + 2*m*(c*x**2)**(3/2)) + 2*a*b*m**2*x**2*(d*x)**m/(m**3*(c*x**2)**(3/2) - 3*m**2*(c*x
**2)**(3/2) + 2*m*(c*x**2)**(3/2)) - 4*a*b*m*x**2*(d*x)**m/(m**3*(c*x**2)**(3/2) - 3*m**2*(c*x**2)**(3/2) + 2*
m*(c*x**2)**(3/2)) + b**2*m**2*x**3*(d*x)**m/(m**3*(c*x**2)**(3/2) - 3*m**2*(c*x**2)**(3/2) + 2*m*(c*x**2)**(3
/2)) - 3*b**2*m*x**3*(d*x)**m/(m**3*(c*x**2)**(3/2) - 3*m**2*(c*x**2)**(3/2) + 2*m*(c*x**2)**(3/2)) + 2*b**2*x
**3*(d*x)**m/(m**3*(c*x**2)**(3/2) - 3*m**2*(c*x**2)**(3/2) + 2*m*(c*x**2)**(3/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.63 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {b^{2} d^{m} x^{m}}{c^{\frac {3}{2}} m} + \frac {2 \, a b d^{m} x^{m}}{c^{\frac {3}{2}} {\left (m - 1\right )} x} + \frac {a^{2} d^{m} x^{m}}{c^{\frac {3}{2}} {\left (m - 2\right )} x^{2}} \]

[In]

integrate((d*x)^m*(b*x+a)^2/(c*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{2} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((d*x)^m*(b*x+a)^2/(c*x^2)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.71 \[ \int \frac {(d x)^m (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {a^2\,{\left (d\,x\right )}^m}{c\,x\,\sqrt {c\,x^2}\,\left (m-2\right )}+\frac {b\,{\left (d\,x\right )}^m\,\left (2\,a\,m-b\,x+b\,m\,x\right )}{c\,m\,\sqrt {c\,x^2}\,\left (m-1\right )} \]

[In]

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

[Out]

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