3.62 \(\int (c x)^m \left (b x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=29 \[ \frac{b \sqrt{b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]

[Out]

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

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Rubi [A]  time = 0.0280846, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b \sqrt{b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(b*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.05319, size = 27, normalized size = 0.93 \[ \frac{b x^{- m} x^{m + 4} \sqrt{b x^{2}} \left (c x\right )^{m}}{x \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00882833, size = 21, normalized size = 0.72 \[ \frac{x \left (b x^2\right )^{3/2} (c x)^m}{m+4} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x)^m*(b*x^2)^(3/2))/(4 + m)

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Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \[{\frac{x \left ( cx \right ) ^{m}}{4+m} \left ( b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(b*x^2)^(3/2),x)

[Out]

x/(4+m)*(c*x)^m*(b*x^2)^(3/2)

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Maxima [A]  time = 1.45965, size = 24, normalized size = 0.83 \[ \frac{b^{\frac{3}{2}} c^{m} x^{4} x^{m}}{m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^(3/2)*c^m*x^4*x^m/(m + 4)

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Fricas [A]  time = 0.269558, size = 30, normalized size = 1.03 \[ \frac{\sqrt{b x^{2}} \left (c x\right )^{m} b x^{3}}{m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2)*(c*x)^m*b*x^3/(m + 4)

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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((c*x)**m*(b*x**2)**(3/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [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((b*x^2)^(3/2)*(c*x)^m,x, algorithm="giac")

[Out]

Exception raised: TypeError