3.926 \(\int \frac{\sqrt{c x^2} (a+b x)^n}{x} \, dx\)

Optimal. Leaf size=30 \[ \frac{\sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

[Out]

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

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Rubi [A]  time = 0.0172442, antiderivative size = 30, 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 \[ \frac{\sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.2308, size = 22, normalized size = 0.73 \[ \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b x \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0179754, size = 29, normalized size = 0.97 \[ \frac{c x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 29, normalized size = 1. \[{\frac{ \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ) x}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(c*x^2)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 1.35338, size = 38, normalized size = 1.27 \[ \frac{{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225717, size = 41, normalized size = 1.37 \[ \frac{\sqrt{c x^{2}}{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.203946, size = 57, normalized size = 1.9 \[ -\sqrt{c}{\left (\frac{a^{n + 1}{\rm sign}\left (x\right )}{b n + b} - \frac{{\left (b x + a\right )}^{n + 1}{\rm sign}\left (x\right )}{b{\left (n + 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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