3.974 \(\int \frac{(d x)^m (a+b x)}{\left (c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{a d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0800258, antiderivative size = 67, 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 \[ -\frac{a d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((d*x)^m*(a + b*x))/(c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.3277, size = 58, normalized size = 0.87 \[ - \frac{a d^{4} \sqrt{c x^{2}} \left (d x\right )^{m - 4}}{c^{3} x \left (- m + 4\right )} - \frac{b d^{3} \sqrt{c x^{2}} \left (d x\right )^{m - 3}}{c^{3} x \left (- m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b*x+a)/(c*x**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0378204, size = 32, normalized size = 0.48 \[ \frac{x (d x)^m \left (\frac{a}{m-4}+\frac{b x}{m-3}\right )}{\left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d*x)^m*(a + b*x))/(c*x^2)^(5/2),x]

[Out]

(x*(d*x)^m*(a/(-4 + m) + (b*x)/(-3 + m)))/(c*x^2)^(5/2)

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Maple [A]  time = 0.005, size = 40, normalized size = 0.6 \[{\frac{ \left ( bmx+am-4\,bx-3\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( -3+m \right ) \left ( -4+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b*x+a)/(c*x^2)^(5/2),x)

[Out]

x*(b*m*x+a*m-4*b*x-3*a)*(d*x)^m/(-3+m)/(-4+m)/(c*x^2)^(5/2)

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Maxima [A]  time = 1.34985, size = 53, normalized size = 0.79 \[ \frac{b d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 3\right )} x^{3}} + \frac{a d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 4\right )} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x)^m/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

b*d^m*x^m/(c^(5/2)*(m - 3)*x^3) + a*d^m*x^m/(c^(5/2)*(m - 4)*x^4)

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Fricas [A]  time = 0.240987, size = 72, normalized size = 1.07 \[ \frac{\sqrt{c x^{2}}{\left (a m +{\left (b m - 4 \, b\right )} x - 3 \, a\right )} \left (d x\right )^{m}}{{\left (c^{3} m^{2} - 7 \, c^{3} m + 12 \, c^{3}\right )} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x)^m/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

sqrt(c*x^2)*(a*m + (b*m - 4*b)*x - 3*a)*(d*x)^m/((c^3*m^2 - 7*c^3*m + 12*c^3)*x^
5)

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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)**(5/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x)^m/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)*(d*x)^m/(c*x^2)^(5/2), x)