3.515 \(\int (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx\)

Optimal. Leaf size=108 \[ \frac{b^2 x^{n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left (a b+b^2 x^n\right )}+\frac{a (d x)^{m+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.103808, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{b^2 x^{n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left (a b+b^2 x^n\right )}+\frac{a (d x)^{m+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.4122, size = 97, normalized size = 0.9 \[ \frac{2 a b n \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{d \left (m + 1\right ) \left (2 a b + 2 b^{2} x^{n}\right ) \left (m + n + 1\right )} + \frac{\left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{d \left (m + n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0497311, size = 55, normalized size = 0.51 \[ \frac{x (d x)^m \sqrt{\left (a+b x^n\right )^2} \left (a (m+n+1)+b (m+1) x^n\right )}{(m+1) (m+n+1) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.064, size = 132, normalized size = 1.2 \[{\frac{x \left ( mb{x}^{n}+am+an+b{x}^{n}+a \right ) }{ \left ( a+b{x}^{n} \right ) \left ( 1+m \right ) \left ( 1+m+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{{\rm e}^{-{\frac{m \left ( i\pi \, \left ({\it csgn} \left ( idx \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) -i\pi \, \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) +i\pi \,{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) -2\,\ln \left ( x \right ) -2\,\ln \left ( d \right ) \right ) }{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((a+b*x^n)^2)^(1/2)/(a+b*x^n)*x*(m*b*x^n+a*m+a*n+b*x^n+a)/(1+m)/(1+m+n)*exp(-1/2
*m*(I*Pi*csgn(I*d*x)^3-I*Pi*csgn(I*d*x)^2*csgn(I*d)-I*Pi*csgn(I*d*x)^2*csgn(I*x)
+I*Pi*csgn(I*d*x)*csgn(I*d)*csgn(I*x)-2*ln(x)-2*ln(d)))

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Maxima [A]  time = 0.76477, size = 63, normalized size = 0.58 \[ \frac{a d^{m}{\left (m + n + 1\right )} x x^{m} + b d^{m}{\left (m + 1\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m^{2} + m{\left (n + 2\right )} + n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*d^m*(m + n + 1)*x*x^m + b*d^m*(m + 1)*x*e^(m*log(x) + n*log(x)))/(m^2 + m*(n
+ 2) + n + 1)

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Fricas [A]  time = 0.274498, size = 77, normalized size = 0.71 \[ \frac{{\left (b m + b\right )} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} +{\left (a m + a n + a\right )} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{2} +{\left (m + 1\right )} n + 2 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*m + b)*x*x^n*e^(m*log(d) + m*log(x)) + (a*m + a*n + a)*x*e^(m*log(d) + m*log
(x)))/(m^2 + (m + 1)*n + 2*m + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{\left (a + b x^{n}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)

[Out]

Integral((d*x)**m*sqrt((a + b*x**n)**2), x)

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GIAC/XCAS [A]  time = 0.285689, size = 236, normalized size = 2.19 \[ \frac{b m x e^{\left (m{\rm ln}\left (d\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + a m x e^{\left (m{\rm ln}\left (d\right ) + m{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + b m x e^{\left (m{\rm ln}\left (d\right ) + m{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + a n x e^{\left (m{\rm ln}\left (d\right ) + m{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + b x e^{\left (m{\rm ln}\left (d\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + a x e^{\left (m{\rm ln}\left (d\right ) + m{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + b x e^{\left (m{\rm ln}\left (d\right ) + m{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right )}{m^{2} + m n + 2 \, m + n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*m*x*e^(m*ln(d) + m*ln(x) + n*ln(x))*sign(b*x^n + a) + a*m*x*e^(m*ln(d) + m*ln
(x))*sign(b*x^n + a) + b*m*x*e^(m*ln(d) + m*ln(x))*sign(b*x^n + a) + a*n*x*e^(m*
ln(d) + m*ln(x))*sign(b*x^n + a) + b*x*e^(m*ln(d) + m*ln(x) + n*ln(x))*sign(b*x^
n + a) + a*x*e^(m*ln(d) + m*ln(x))*sign(b*x^n + a) + b*x*e^(m*ln(d) + m*ln(x))*s
ign(b*x^n + a))/(m^2 + m*n + 2*m + n + 1)