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

Optimal. Leaf size=105 \[ -\frac{a^2 d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{2 a b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}}-\frac{b^2 d^2 x (d x)^{m-2}}{c^2 (2-m) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.126941, antiderivative size = 105, 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 \[ -\frac{a^2 d^4 x (d x)^{m-4}}{c^2 (4-m) \sqrt{c x^2}}-\frac{2 a b d^3 x (d x)^{m-3}}{c^2 (3-m) \sqrt{c x^2}}-\frac{b^2 d^2 x (d x)^{m-2}}{c^2 (2-m) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.674, size = 94, normalized size = 0.9 \[ - \frac{a^{2} d^{4} \sqrt{c x^{2}} \left (d x\right )^{m - 4}}{c^{3} x \left (- m + 4\right )} - \frac{2 a b d^{3} \sqrt{c x^{2}} \left (d x\right )^{m - 3}}{c^{3} x \left (- m + 3\right )} - \frac{b^{2} d^{2} \sqrt{c x^{2}} \left (d x\right )^{m - 2}}{c^{3} x \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0655092, size = 48, normalized size = 0.46 \[ \frac{x (d x)^m \left (\frac{a^2}{m-4}+\frac{2 a b x}{m-3}+\frac{b^2 x^2}{m-2}\right )}{\left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 95, normalized size = 0.9 \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x-7\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}-12\,abmx+12\,{b}^{2}{x}^{2}-5\,{a}^{2}m+16\,abx+6\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( -2+m \right ) \left ( -3+m \right ) \left ( -4+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x-7*b^2*m*x^2+a^2*m^2-12*a*b*m*x+12*b^2*x^2-5*a^2*m+16*
a*b*x+6*a^2)*(d*x)^m/(-2+m)/(-3+m)/(-4+m)/(c*x^2)^(5/2)

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Maxima [A]  time = 1.3593, size = 86, normalized size = 0.82 \[ \frac{b^{2} d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 2\right )} x^{2}} + \frac{2 \, a b d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 3\right )} x^{3}} + \frac{a^{2} d^{m} x^{m}}{c^{\frac{5}{2}}{\left (m - 4\right )} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232356, size = 143, normalized size = 1.36 \[ \frac{{\left (a^{2} m^{2} - 5 \, a^{2} m +{\left (b^{2} m^{2} - 7 \, b^{2} m + 12 \, b^{2}\right )} x^{2} + 6 \, a^{2} + 2 \,{\left (a b m^{2} - 6 \, a b m + 8 \, a b\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{{\left (c^{3} m^{3} - 9 \, c^{3} m^{2} + 26 \, c^{3} m - 24 \, c^{3}\right )} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^2*m^2 - 5*a^2*m + (b^2*m^2 - 7*b^2*m + 12*b^2)*x^2 + 6*a^2 + 2*(a*b*m^2 - 6*a
*b*m + 8*a*b)*x)*sqrt(c*x^2)*(d*x)^m/((c^3*m^3 - 9*c^3*m^2 + 26*c^3*m - 24*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)**2/(c*x**2)**(5/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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