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

Optimal. Leaf size=31 \[ \frac{c \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

[Out]

(c*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b*(1 + n)*x)

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Rubi [A]  time = 0.0191996, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{c \sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b*(1 + n)*x)

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Rubi in Sympy [A]  time = 12.3545, size = 24, normalized size = 0.77 \[ \frac{c \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*sqrt(c*x**2)*(a + b*x)**(n + 1)/(b*x*(n + 1))

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Mathematica [A]  time = 0.0202143, size = 30, normalized size = 0.97 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^{n+1}}{b (n+1) x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 29, normalized size = 0.9 \[{\frac{ \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ){x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36028, size = 38, normalized size = 1.23 \[ \frac{{\left (b c^{\frac{3}{2}} x + a c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23168, size = 45, normalized size = 1.45 \[ \frac{{\left (b c x + a c\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*c*x + a*c)*sqrt(c*x^2)*(b*x + a)^n/((b*n + b)*x)

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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**2)**(3/2)*(b*x+a)**n/x**3,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.206723, size = 57, normalized size = 1.84 \[ -c^{\frac{3}{2}}{\left (\frac{a^{n + 1}{\rm sign}\left (x\right )}{b n + b} - \frac{{\left (b x + a\right )}^{n + 1}{\rm sign}\left (x\right )}{b{\left (n + 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c^(3/2)*(a^(n + 1)*sign(x)/(b*n + b) - (b*x + a)^(n + 1)*sign(x)/(b*(n + 1)))