3.2822 \(\int \frac{1}{\left (\frac{c}{(a+b x)^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{(a+b x)^3}{4 b c \sqrt{\frac{c}{(a+b x)^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0260905, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a+b x)^3}{4 b c \sqrt{\frac{c}{(a+b x)^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^2)^(-3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.56119, size = 24, normalized size = 0.8 \[ \frac{\sqrt{\frac{c}{\left (a + b x\right )^{2}}} \left (a + b x\right )^{5}}{4 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c/(b*x+a)**2)**(3/2),x)

[Out]

sqrt(c/(a + b*x)**2)*(a + b*x)**5/(4*b*c**2)

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Mathematica [A]  time = 0.0248684, size = 25, normalized size = 0.83 \[ \frac{a+b x}{4 b \left (\frac{c}{(a+b x)^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)/(4*b*(c/(a + b*x)^2)^(3/2))

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Maple [A]  time = 0.004, size = 51, normalized size = 1.7 \[{\frac{x \left ({b}^{3}{x}^{3}+4\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+4\,{a}^{3} \right ) }{4\, \left ( bx+a \right ) ^{3}} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a)^2)^(3/2),x)

[Out]

1/4*x*(b^3*x^3+4*a*b^2*x^2+6*a^2*b*x+4*a^3)/(b*x+a)^3/(c/(b*x+a)^2)^(3/2)

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Maxima [A]  time = 1.39062, size = 50, normalized size = 1.67 \[ \frac{b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x}{4 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)/c^(3/2)

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Fricas [A]  time = 0.215979, size = 95, normalized size = 3.17 \[ \frac{{\left (b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 4 \, a^{4} x\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 4*a^4*x)*sqrt(c/(b^
2*x^2 + 2*a*b*x + a^2))/c^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\frac{c}{\left (a + b x\right )^{2}}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c/(b*x+a)**2)**(3/2),x)

[Out]

Integral((c/(a + b*x)**2)**(-3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\frac{c}{{\left (b x + a\right )}^{2}}\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c/(b*x + a)^2)^(-3/2), x)