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

Optimal. Leaf size=63 \[ \frac{\sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]

[Out]

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

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Rubi [A]  time = 0.0447093, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*x^2]*(a + b*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.2611, size = 51, normalized size = 0.81 \[ - \frac{a \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{2} x \left (n + 1\right )} + \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{2} x \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0408042, size = 44, normalized size = 0.7 \[ \frac{c x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 46, normalized size = 0.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -bxn-bx+a \right ) }{x{b}^{2} \left ({n}^{2}+3\,n+2 \right ) }\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(c*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.36443, size = 69, normalized size = 1.1 \[ \frac{{\left (b^{2} \sqrt{c}{\left (n + 1\right )} x^{2} + a b \sqrt{c} n x - a^{2} \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225294, size = 85, normalized size = 1.35 \[ \frac{{\left (a b n x +{\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.208505, size = 174, normalized size = 2.76 \[{\left (\frac{a^{2} e^{\left (n{\rm ln}\left (a\right )\right )}{\rm sign}\left (x\right )}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} + \frac{b^{2} n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + a b n x e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + b^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - a^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right )}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^2*e^(n*ln(a))*sign(x)/(b^2*n^2 + 3*b^2*n + 2*b^2) + (b^2*n*x^2*e^(n*ln(b*x +
a))*sign(x) + a*b*n*x*e^(n*ln(b*x + a))*sign(x) + b^2*x^2*e^(n*ln(b*x + a))*sign
(x) - a^2*e^(n*ln(b*x + a))*sign(x))/(b^2*n^2 + 3*b^2*n + 2*b^2))*sqrt(c)