3.119 \(\int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx\)

Optimal. Leaf size=205 \[ \frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^3\right )}+\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left (a+b x^3\right )}+\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]

[Out]

(a^3*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d*(1 + m)*(a + b*x^3)) + (3
*a^2*b*(d*x)^(4 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^4*(4 + m)*(a + b*x^3))
+ (3*a*b^2*(d*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^7*(7 + m)*(a + b*x^
3)) + (b^3*(d*x)^(10 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^10*(10 + m)*(a + b
*x^3))

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Rubi [A]  time = 0.199124, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^3\right )}+\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left (a+b x^3\right )}+\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

(a^3*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d*(1 + m)*(a + b*x^3)) + (3
*a^2*b*(d*x)^(4 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^4*(4 + m)*(a + b*x^3))
+ (3*a*b^2*(d*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^7*(7 + m)*(a + b*x^
3)) + (b^3*(d*x)^(10 + m)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(d^10*(10 + m)*(a + b
*x^3))

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Rubi in Sympy [A]  time = 30.6307, size = 182, normalized size = 0.89 \[ \frac{162 a^{3} \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{d \left (a + b x^{3}\right ) \left (m + 1\right ) \left (m + 4\right ) \left (m + 7\right ) \left (m + 10\right )} + \frac{54 a^{2} \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{d \left (m + 4\right ) \left (m + 7\right ) \left (m + 10\right )} + \frac{9 a \left (d x\right )^{m + 1} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{d \left (m + 7\right ) \left (m + 10\right )} + \frac{\left (d x\right )^{m + 1} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{d \left (m + 10\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

162*a**3*(d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(d*(a + b*x**3)*(m +
1)*(m + 4)*(m + 7)*(m + 10)) + 54*a**2*(d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**3 +
b**2*x**6)/(d*(m + 4)*(m + 7)*(m + 10)) + 9*a*(d*x)**(m + 1)*(a + b*x**3)*sqrt(a
**2 + 2*a*b*x**3 + b**2*x**6)/(d*(m + 7)*(m + 10)) + (d*x)**(m + 1)*(a**2 + 2*a*
b*x**3 + b**2*x**6)**(3/2)/(d*(m + 10))

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Mathematica [A]  time = 0.0704961, size = 79, normalized size = 0.39 \[ \frac{\left (\left (a+b x^3\right )^2\right )^{3/2} (d x)^m \left (\frac{a^3 x}{m+1}+\frac{3 a^2 b x^4}{m+4}+\frac{3 a b^2 x^7}{m+7}+\frac{b^3 x^{10}}{m+10}\right )}{\left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2),x]

[Out]

((d*x)^m*((a + b*x^3)^2)^(3/2)*((a^3*x)/(1 + m) + (3*a^2*b*x^4)/(4 + m) + (3*a*b
^2*x^7)/(7 + m) + (b^3*x^10)/(10 + m)))/(a + b*x^3)^3

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Maple [A]  time = 0.009, size = 199, normalized size = 1. \[{\frac{ \left ({b}^{3}{m}^{3}{x}^{9}+12\,{b}^{3}{m}^{2}{x}^{9}+39\,{b}^{3}m{x}^{9}+3\,a{b}^{2}{m}^{3}{x}^{6}+28\,{b}^{3}{x}^{9}+45\,a{b}^{2}{m}^{2}{x}^{6}+162\,a{b}^{2}m{x}^{6}+3\,{a}^{2}b{m}^{3}{x}^{3}+120\,a{x}^{6}{b}^{2}+54\,{a}^{2}b{m}^{2}{x}^{3}+261\,{a}^{2}bm{x}^{3}+{a}^{3}{m}^{3}+210\,{x}^{3}{a}^{2}b+21\,{a}^{3}{m}^{2}+138\,{a}^{3}m+280\,{a}^{3} \right ) x \left ( dx \right ) ^{m}}{ \left ( 10+m \right ) \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

x*(b^3*m^3*x^9+12*b^3*m^2*x^9+39*b^3*m*x^9+3*a*b^2*m^3*x^6+28*b^3*x^9+45*a*b^2*m
^2*x^6+162*a*b^2*m*x^6+3*a^2*b*m^3*x^3+120*a*b^2*x^6+54*a^2*b*m^2*x^3+261*a^2*b*
m*x^3+a^3*m^3+210*a^2*b*x^3+21*a^3*m^2+138*a^3*m+280*a^3)*(d*x)^m*((b*x^3+a)^2)^
(3/2)/(10+m)/(7+m)/(4+m)/(1+m)/(b*x^3+a)^3

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Maxima [A]  time = 0.79959, size = 161, normalized size = 0.79 \[ \frac{{\left ({\left (m^{3} + 12 \, m^{2} + 39 \, m + 28\right )} b^{3} d^{m} x^{10} + 3 \,{\left (m^{3} + 15 \, m^{2} + 54 \, m + 40\right )} a b^{2} d^{m} x^{7} + 3 \,{\left (m^{3} + 18 \, m^{2} + 87 \, m + 70\right )} a^{2} b d^{m} x^{4} +{\left (m^{3} + 21 \, m^{2} + 138 \, m + 280\right )} a^{3} d^{m} x\right )} x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*(d*x)^m,x, algorithm="maxima")

[Out]

((m^3 + 12*m^2 + 39*m + 28)*b^3*d^m*x^10 + 3*(m^3 + 15*m^2 + 54*m + 40)*a*b^2*d^
m*x^7 + 3*(m^3 + 18*m^2 + 87*m + 70)*a^2*b*d^m*x^4 + (m^3 + 21*m^2 + 138*m + 280
)*a^3*d^m*x)*x^m/(m^4 + 22*m^3 + 159*m^2 + 418*m + 280)

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Fricas [A]  time = 0.267396, size = 215, normalized size = 1.05 \[ \frac{{\left ({\left (b^{3} m^{3} + 12 \, b^{3} m^{2} + 39 \, b^{3} m + 28 \, b^{3}\right )} x^{10} + 3 \,{\left (a b^{2} m^{3} + 15 \, a b^{2} m^{2} + 54 \, a b^{2} m + 40 \, a b^{2}\right )} x^{7} + 3 \,{\left (a^{2} b m^{3} + 18 \, a^{2} b m^{2} + 87 \, a^{2} b m + 70 \, a^{2} b\right )} x^{4} +{\left (a^{3} m^{3} + 21 \, a^{3} m^{2} + 138 \, a^{3} m + 280 \, a^{3}\right )} x\right )} \left (d x\right )^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*(d*x)^m,x, algorithm="fricas")

[Out]

((b^3*m^3 + 12*b^3*m^2 + 39*b^3*m + 28*b^3)*x^10 + 3*(a*b^2*m^3 + 15*a*b^2*m^2 +
54*a*b^2*m + 40*a*b^2)*x^7 + 3*(a^2*b*m^3 + 18*a^2*b*m^2 + 87*a^2*b*m + 70*a^2*
b)*x^4 + (a^3*m^3 + 21*a^3*m^2 + 138*a^3*m + 280*a^3)*x)*(d*x)^m/(m^4 + 22*m^3 +
159*m^2 + 418*m + 280)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Integral((d*x)**m*((a + b*x**3)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.301742, size = 562, normalized size = 2.74 \[ \frac{b^{3} m^{3} x^{10} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 12 \, b^{3} m^{2} x^{10} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 39 \, b^{3} m x^{10} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 3 \, a b^{2} m^{3} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 28 \, b^{3} x^{10} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 45 \, a b^{2} m^{2} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 162 \, a b^{2} m x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 3 \, a^{2} b m^{3} x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 120 \, a b^{2} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 54 \, a^{2} b m^{2} x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 261 \, a^{2} b m x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + a^{3} m^{3} x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 210 \, a^{2} b x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 21 \, a^{3} m^{2} x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 138 \, a^{3} m x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right ) + 280 \, a^{3} x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{3} + a\right )}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*(d*x)^m,x, algorithm="giac")

[Out]

(b^3*m^3*x^10*e^(m*ln(d*x))*sign(b*x^3 + a) + 12*b^3*m^2*x^10*e^(m*ln(d*x))*sign
(b*x^3 + a) + 39*b^3*m*x^10*e^(m*ln(d*x))*sign(b*x^3 + a) + 3*a*b^2*m^3*x^7*e^(m
*ln(d*x))*sign(b*x^3 + a) + 28*b^3*x^10*e^(m*ln(d*x))*sign(b*x^3 + a) + 45*a*b^2
*m^2*x^7*e^(m*ln(d*x))*sign(b*x^3 + a) + 162*a*b^2*m*x^7*e^(m*ln(d*x))*sign(b*x^
3 + a) + 3*a^2*b*m^3*x^4*e^(m*ln(d*x))*sign(b*x^3 + a) + 120*a*b^2*x^7*e^(m*ln(d
*x))*sign(b*x^3 + a) + 54*a^2*b*m^2*x^4*e^(m*ln(d*x))*sign(b*x^3 + a) + 261*a^2*
b*m*x^4*e^(m*ln(d*x))*sign(b*x^3 + a) + a^3*m^3*x*e^(m*ln(d*x))*sign(b*x^3 + a)
+ 210*a^2*b*x^4*e^(m*ln(d*x))*sign(b*x^3 + a) + 21*a^3*m^2*x*e^(m*ln(d*x))*sign(
b*x^3 + a) + 138*a^3*m*x*e^(m*ln(d*x))*sign(b*x^3 + a) + 280*a^3*x*e^(m*ln(d*x))
*sign(b*x^3 + a))/(m^4 + 22*m^3 + 159*m^2 + 418*m + 280)