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

Optimal. Leaf size=93 \[ -\frac {a^2 d^2 x (d x)^{-2+m}}{c (2-m) \sqrt {c x^2}}-\frac {2 a b d x (d x)^{-1+m}}{c (1-m) \sqrt {c x^2}}+\frac {b^2 x (d x)^m}{c m \sqrt {c x^2}} \]

[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)

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Rubi [A]
time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, 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, 45} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.04, size = 62, normalized size = 0.67 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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)
)

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.12, size = 83, normalized size = 0.89

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

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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Maxima [A]
time = 0.27, size = 59, normalized size = 0.63 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.30, size = 92, normalized size = 0.99 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \int \frac {\left (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx & \text {for}\: m = 0 \\d \left (\int \frac {a^{2} x}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx + \int \frac {b^{2} x^{3}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 a b x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx\right ) & \text {for}\: m = 1 \\d^{2} \left (\int \frac {a^{2} x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx + \int \frac {b^{2} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 a b x^{3}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((Integral((a + b*x)**2/(c*x**2)**(3/2), x), Eq(m, 0)), (d*(Integral(a**2*x/(c*x**2)**(3/2), x) + Int
egral(b**2*x**3/(c*x**2)**(3/2), x) + Integral(2*a*b*x**2/(c*x**2)**(3/2), x)), Eq(m, 1)), (d**2*(Integral(a**
2*x**2/(c*x**2)**(3/2), x) + Integral(b**2*x**4/(c*x**2)**(3/2), x) + Integral(2*a*b*x**3/(c*x**2)**(3/2), x))
, 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))

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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Mupad [B]
time = 0.32, size = 66, normalized size = 0.71 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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