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

Optimal. Leaf size=65 \[ -\frac{a d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0792115, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{a d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.742, size = 56, normalized size = 0.86 \[ - \frac{a d^{2} \sqrt{c x^{2}} \left (d x\right )^{m - 2}}{c^{2} x \left (- m + 2\right )} - \frac{b d \sqrt{c x^{2}} \left (d x\right )^{m - 1}}{c^{2} x \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.03946, size = 32, normalized size = 0.49 \[ \frac{x (d x)^m \left (\frac{a}{m-2}+\frac{b x}{m-1}\right )}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 40, normalized size = 0.6 \[{\frac{ \left ( bmx+am-2\,bx-a \right ) x \left ( dx \right ) ^{m}}{ \left ( -1+m \right ) \left ( -2+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.42635, size = 53, normalized size = 0.82 \[ \frac{b d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 1\right )} x} + \frac{a d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23291, size = 72, normalized size = 1.11 \[ \frac{\sqrt{c x^{2}}{\left (a m +{\left (b m - 2 \, b\right )} x - a\right )} \left (d x\right )^{m}}{{\left (c^{2} m^{2} - 3 \, c^{2} m + 2 \, c^{2}\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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