3.792 \(\int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=205 \[ \frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]

[Out]

(a^3*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*(1 + m)*(a + b*x^2)) + (3
*a^2*b*(d*x)^(3 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(3 + m)*(a + b*x^2))
+ (3*a*b^2*(d*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^5*(5 + m)*(a + b*x^
2)) + (b^3*(d*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^7*(7 + m)*(a + b*x^
2))

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Rubi [A]  time = 0.219708, 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 \[ \frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(a^3*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*(1 + m)*(a + b*x^2)) + (3
*a^2*b*(d*x)^(3 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^3*(3 + m)*(a + b*x^2))
+ (3*a*b^2*(d*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^5*(5 + m)*(a + b*x^
2)) + (b^3*(d*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^7*(7 + m)*(a + b*x^
2))

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Rubi in Sympy [A]  time = 29.9014, size = 182, normalized size = 0.89 \[ \frac{48 a^{3} \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{d \left (a + b x^{2}\right ) \left (m + 1\right ) \left (m + 3\right ) \left (m + 5\right ) \left (m + 7\right )} + \frac{24 a^{2} \left (d x\right )^{m + 1} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{d \left (m + 3\right ) \left (m + 5\right ) \left (m + 7\right )} + \frac{6 a \left (d x\right )^{m + 1} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{d \left (m + 5\right ) \left (m + 7\right )} + \frac{\left (d x\right )^{m + 1} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{d \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

48*a**3*(d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(d*(a + b*x**2)*(m +
1)*(m + 3)*(m + 5)*(m + 7)) + 24*a**2*(d*x)**(m + 1)*sqrt(a**2 + 2*a*b*x**2 + b*
*2*x**4)/(d*(m + 3)*(m + 5)*(m + 7)) + 6*a*(d*x)**(m + 1)*(a + b*x**2)*sqrt(a**2
+ 2*a*b*x**2 + b**2*x**4)/(d*(m + 5)*(m + 7)) + (d*x)**(m + 1)*(a**2 + 2*a*b*x*
*2 + b**2*x**4)**(3/2)/(d*(m + 7))

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Mathematica [A]  time = 0.0721606, size = 79, normalized size = 0.39 \[ \frac{\left (\left (a+b x^2\right )^2\right )^{3/2} (d x)^m \left (\frac{a^3 x}{m+1}+\frac{3 a^2 b x^3}{m+3}+\frac{3 a b^2 x^5}{m+5}+\frac{b^3 x^7}{m+7}\right )}{\left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

((d*x)^m*((a + b*x^2)^2)^(3/2)*((a^3*x)/(1 + m) + (3*a^2*b*x^3)/(3 + m) + (3*a*b
^2*x^5)/(5 + m) + (b^3*x^7)/(7 + m)))/(a + b*x^2)^3

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Maple [A]  time = 0.009, size = 199, normalized size = 1. \[{\frac{ \left ({b}^{3}{m}^{3}{x}^{6}+9\,{b}^{3}{m}^{2}{x}^{6}+3\,a{b}^{2}{m}^{3}{x}^{4}+23\,{b}^{3}m{x}^{6}+33\,a{b}^{2}{m}^{2}{x}^{4}+15\,{b}^{3}{x}^{6}+3\,{a}^{2}b{m}^{3}{x}^{2}+93\,a{b}^{2}m{x}^{4}+39\,{a}^{2}b{m}^{2}{x}^{2}+63\,a{x}^{4}{b}^{2}+{a}^{3}{m}^{3}+141\,{a}^{2}bm{x}^{2}+15\,{a}^{3}{m}^{2}+105\,{a}^{2}b{x}^{2}+71\,{a}^{3}m+105\,{a}^{3} \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

x*(b^3*m^3*x^6+9*b^3*m^2*x^6+3*a*b^2*m^3*x^4+23*b^3*m*x^6+33*a*b^2*m^2*x^4+15*b^
3*x^6+3*a^2*b*m^3*x^2+93*a*b^2*m*x^4+39*a^2*b*m^2*x^2+63*a*b^2*x^4+a^3*m^3+141*a
^2*b*m*x^2+15*a^3*m^2+105*a^2*b*x^2+71*a^3*m+105*a^3)*(d*x)^m*((b*x^2+a)^2)^(3/2
)/(7+m)/(5+m)/(3+m)/(1+m)/(b*x^2+a)^3

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Maxima [A]  time = 0.691084, size = 161, normalized size = 0.79 \[ \frac{{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} d^{m} x^{7} + 3 \,{\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} d^{m} x^{5} + 3 \,{\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b d^{m} x^{3} +{\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} d^{m} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*(d*x)^m,x, algorithm="maxima")

[Out]

((m^3 + 9*m^2 + 23*m + 15)*b^3*d^m*x^7 + 3*(m^3 + 11*m^2 + 31*m + 21)*a*b^2*d^m*
x^5 + 3*(m^3 + 13*m^2 + 47*m + 35)*a^2*b*d^m*x^3 + (m^3 + 15*m^2 + 71*m + 105)*a
^3*d^m*x)*x^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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Fricas [A]  time = 0.287, size = 215, normalized size = 1.05 \[ \frac{{\left ({\left (b^{3} m^{3} + 9 \, b^{3} m^{2} + 23 \, b^{3} m + 15 \, b^{3}\right )} x^{7} + 3 \,{\left (a b^{2} m^{3} + 11 \, a b^{2} m^{2} + 31 \, a b^{2} m + 21 \, a b^{2}\right )} x^{5} + 3 \,{\left (a^{2} b m^{3} + 13 \, a^{2} b m^{2} + 47 \, a^{2} b m + 35 \, a^{2} b\right )} x^{3} +{\left (a^{3} m^{3} + 15 \, a^{3} m^{2} + 71 \, a^{3} m + 105 \, a^{3}\right )} x\right )} \left (d x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*(d*x)^m,x, algorithm="fricas")

[Out]

((b^3*m^3 + 9*b^3*m^2 + 23*b^3*m + 15*b^3)*x^7 + 3*(a*b^2*m^3 + 11*a*b^2*m^2 + 3
1*a*b^2*m + 21*a*b^2)*x^5 + 3*(a^2*b*m^3 + 13*a^2*b*m^2 + 47*a^2*b*m + 35*a^2*b)
*x^3 + (a^3*m^3 + 15*a^3*m^2 + 71*a^3*m + 105*a^3)*x)*(d*x)^m/(m^4 + 16*m^3 + 86
*m^2 + 176*m + 105)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.27468, size = 562, normalized size = 2.74 \[ \frac{b^{3} m^{3} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 9 \, b^{3} m^{2} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 3 \, a b^{2} m^{3} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 23 \, b^{3} m x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 33 \, a b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 15 \, b^{3} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 3 \, a^{2} b m^{3} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 93 \, a b^{2} m x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 39 \, a^{2} b m^{2} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 63 \, a b^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + a^{3} m^{3} x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 141 \, a^{2} b m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 15 \, a^{3} m^{2} x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 105 \, a^{2} b x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 71 \, a^{3} m x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right ) + 105 \, a^{3} x e^{\left (m{\rm ln}\left (d x\right )\right )}{\rm sign}\left (b x^{2} + a\right )}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*(d*x)^m,x, algorithm="giac")

[Out]

(b^3*m^3*x^7*e^(m*ln(d*x))*sign(b*x^2 + a) + 9*b^3*m^2*x^7*e^(m*ln(d*x))*sign(b*
x^2 + a) + 3*a*b^2*m^3*x^5*e^(m*ln(d*x))*sign(b*x^2 + a) + 23*b^3*m*x^7*e^(m*ln(
d*x))*sign(b*x^2 + a) + 33*a*b^2*m^2*x^5*e^(m*ln(d*x))*sign(b*x^2 + a) + 15*b^3*
x^7*e^(m*ln(d*x))*sign(b*x^2 + a) + 3*a^2*b*m^3*x^3*e^(m*ln(d*x))*sign(b*x^2 + a
) + 93*a*b^2*m*x^5*e^(m*ln(d*x))*sign(b*x^2 + a) + 39*a^2*b*m^2*x^3*e^(m*ln(d*x)
)*sign(b*x^2 + a) + 63*a*b^2*x^5*e^(m*ln(d*x))*sign(b*x^2 + a) + a^3*m^3*x*e^(m*
ln(d*x))*sign(b*x^2 + a) + 141*a^2*b*m*x^3*e^(m*ln(d*x))*sign(b*x^2 + a) + 15*a^
3*m^2*x*e^(m*ln(d*x))*sign(b*x^2 + a) + 105*a^2*b*x^3*e^(m*ln(d*x))*sign(b*x^2 +
a) + 71*a^3*m*x*e^(m*ln(d*x))*sign(b*x^2 + a) + 105*a^3*x*e^(m*ln(d*x))*sign(b*
x^2 + a))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)