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

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

[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])

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Rubi [A]  time = 0.105703, antiderivative size = 93, 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 \[ -\frac{a^2 d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{2 a b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}}+\frac{b^2 x (d x)^m}{c m \sqrt{c x^2}} \]

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])

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Rubi in Sympy [A]  time = 29.2067, size = 83, normalized size = 0.89 \[ - \frac{a^{2} d^{2} \sqrt{c x^{2}} \left (d x\right )^{m - 2}}{c^{2} x \left (- m + 2\right )} - \frac{2 a b d \sqrt{c x^{2}} \left (d x\right )^{m - 1}}{c^{2} x \left (- m + 1\right )} + \frac{b^{2} \sqrt{c x^{2}} \left (d x\right )^{m}}{c^{2} m x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0552515, size = 50, normalized size = 0.54 \[ \frac{x^3 (d x)^m \left (\frac{a^2}{(m-2) x^2}+\frac{2 a b}{(m-1) x}+\frac{b^2}{m}\right )}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 83, normalized size = 0.9 \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,abx{m}^{2}-3\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}-4\,abxm+2\,{b}^{2}{x}^{2}-{a}^{2}m \right ) x \left ( dx \right ) ^{m}}{m \left ( -1+m \right ) \left ( -2+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)

[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 = 1.38114, size = 80, normalized size = 0.86 \[ \frac{b^{2} d^{m} x^{m}}{c^{\frac{3}{2}} m} + \frac{2 \, a b d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 1\right )} x} + \frac{a^{2} d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*(d*x)^m/(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 = 0.232149, size = 124, normalized size = 1.33 \[ \frac{{\left (a^{2} m^{2} - a^{2} m +{\left (b^{2} m^{2} - 3 \, b^{2} m + 2 \, b^{2}\right )} x^{2} + 2 \,{\left (a b m^{2} - 2 \, a b m\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{{\left (c^{2} m^{3} - 3 \, c^{2} m^{2} + 2 \, c^{2} m\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*(d*x)^m/(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)*sq
rt(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(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{2} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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