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

Optimal. Leaf size=131 \[ \frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}} \]

[Out]

(2*a^6*(d*x)^(5/2))/(5*d) + (4*a^5*b*(d*x)^(9/2))/(3*d^3) + (30*a^4*b^2*(d*x)^(1
3/2))/(13*d^5) + (40*a^3*b^3*(d*x)^(17/2))/(17*d^7) + (10*a^2*b^4*(d*x)^(21/2))/
(7*d^9) + (12*a*b^5*(d*x)^(25/2))/(25*d^11) + (2*b^6*(d*x)^(29/2))/(29*d^13)

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Rubi [A]  time = 0.152198, antiderivative size = 131, 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 \[ \frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^6*(d*x)^(5/2))/(5*d) + (4*a^5*b*(d*x)^(9/2))/(3*d^3) + (30*a^4*b^2*(d*x)^(1
3/2))/(13*d^5) + (40*a^3*b^3*(d*x)^(17/2))/(17*d^7) + (10*a^2*b^4*(d*x)^(21/2))/
(7*d^9) + (12*a*b^5*(d*x)^(25/2))/(25*d^11) + (2*b^6*(d*x)^(29/2))/(29*d^13)

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Rubi in Sympy [A]  time = 36.0061, size = 129, normalized size = 0.98 \[ \frac{2 a^{6} \left (d x\right )^{\frac{5}{2}}}{5 d} + \frac{4 a^{5} b \left (d x\right )^{\frac{9}{2}}}{3 d^{3}} + \frac{30 a^{4} b^{2} \left (d x\right )^{\frac{13}{2}}}{13 d^{5}} + \frac{40 a^{3} b^{3} \left (d x\right )^{\frac{17}{2}}}{17 d^{7}} + \frac{10 a^{2} b^{4} \left (d x\right )^{\frac{21}{2}}}{7 d^{9}} + \frac{12 a b^{5} \left (d x\right )^{\frac{25}{2}}}{25 d^{11}} + \frac{2 b^{6} \left (d x\right )^{\frac{29}{2}}}{29 d^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**6*(d*x)**(5/2)/(5*d) + 4*a**5*b*(d*x)**(9/2)/(3*d**3) + 30*a**4*b**2*(d*x)*
*(13/2)/(13*d**5) + 40*a**3*b**3*(d*x)**(17/2)/(17*d**7) + 10*a**2*b**4*(d*x)**(
21/2)/(7*d**9) + 12*a*b**5*(d*x)**(25/2)/(25*d**11) + 2*b**6*(d*x)**(29/2)/(29*d
**13)

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Mathematica [A]  time = 0.0297783, size = 77, normalized size = 0.59 \[ \frac{2 x (d x)^{3/2} \left (672945 a^6+2243150 a^5 b x^2+3882375 a^4 b^2 x^4+3958500 a^3 b^3 x^6+2403375 a^2 b^4 x^8+807534 a b^5 x^{10}+116025 b^6 x^{12}\right )}{3364725} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*x)^(3/2)*(672945*a^6 + 2243150*a^5*b*x^2 + 3882375*a^4*b^2*x^4 + 3958500
*a^3*b^3*x^6 + 2403375*a^2*b^4*x^8 + 807534*a*b^5*x^10 + 116025*b^6*x^12))/33647
25

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Maple [A]  time = 0.011, size = 74, normalized size = 0.6 \[{\frac{2\,x \left ( 116025\,{b}^{6}{x}^{12}+807534\,a{b}^{5}{x}^{10}+2403375\,{a}^{2}{b}^{4}{x}^{8}+3958500\,{a}^{3}{b}^{3}{x}^{6}+3882375\,{a}^{4}{b}^{2}{x}^{4}+2243150\,{a}^{5}b{x}^{2}+672945\,{a}^{6} \right ) }{3364725} \left ( dx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3364725*x*(116025*b^6*x^12+807534*a*b^5*x^10+2403375*a^2*b^4*x^8+3958500*a^3*b
^3*x^6+3882375*a^4*b^2*x^4+2243150*a^5*b*x^2+672945*a^6)*(d*x)^(3/2)

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Maxima [A]  time = 0.705166, size = 142, normalized size = 1.08 \[ \frac{2 \,{\left (116025 \, \left (d x\right )^{\frac{29}{2}} b^{6} + 807534 \, \left (d x\right )^{\frac{25}{2}} a b^{5} d^{2} + 2403375 \, \left (d x\right )^{\frac{21}{2}} a^{2} b^{4} d^{4} + 3958500 \, \left (d x\right )^{\frac{17}{2}} a^{3} b^{3} d^{6} + 3882375 \, \left (d x\right )^{\frac{13}{2}} a^{4} b^{2} d^{8} + 2243150 \, \left (d x\right )^{\frac{9}{2}} a^{5} b d^{10} + 672945 \, \left (d x\right )^{\frac{5}{2}} a^{6} d^{12}\right )}}{3364725 \, d^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3364725*(116025*(d*x)^(29/2)*b^6 + 807534*(d*x)^(25/2)*a*b^5*d^2 + 2403375*(d*
x)^(21/2)*a^2*b^4*d^4 + 3958500*(d*x)^(17/2)*a^3*b^3*d^6 + 3882375*(d*x)^(13/2)*
a^4*b^2*d^8 + 2243150*(d*x)^(9/2)*a^5*b*d^10 + 672945*(d*x)^(5/2)*a^6*d^12)/d^13

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Fricas [A]  time = 0.256682, size = 111, normalized size = 0.85 \[ \frac{2}{3364725} \,{\left (116025 \, b^{6} d x^{14} + 807534 \, a b^{5} d x^{12} + 2403375 \, a^{2} b^{4} d x^{10} + 3958500 \, a^{3} b^{3} d x^{8} + 3882375 \, a^{4} b^{2} d x^{6} + 2243150 \, a^{5} b d x^{4} + 672945 \, a^{6} d x^{2}\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3364725*(116025*b^6*d*x^14 + 807534*a*b^5*d*x^12 + 2403375*a^2*b^4*d*x^10 + 39
58500*a^3*b^3*d*x^8 + 3882375*a^4*b^2*d*x^6 + 2243150*a^5*b*d*x^4 + 672945*a^6*d
*x^2)*sqrt(d*x)

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Sympy [A]  time = 19.3357, size = 131, normalized size = 1. \[ \frac{2 a^{6} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a^{5} b d^{\frac{3}{2}} x^{\frac{9}{2}}}{3} + \frac{30 a^{4} b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} + \frac{40 a^{3} b^{3} d^{\frac{3}{2}} x^{\frac{17}{2}}}{17} + \frac{10 a^{2} b^{4} d^{\frac{3}{2}} x^{\frac{21}{2}}}{7} + \frac{12 a b^{5} d^{\frac{3}{2}} x^{\frac{25}{2}}}{25} + \frac{2 b^{6} d^{\frac{3}{2}} x^{\frac{29}{2}}}{29} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**6*d**(3/2)*x**(5/2)/5 + 4*a**5*b*d**(3/2)*x**(9/2)/3 + 30*a**4*b**2*d**(3/2
)*x**(13/2)/13 + 40*a**3*b**3*d**(3/2)*x**(17/2)/17 + 10*a**2*b**4*d**(3/2)*x**(
21/2)/7 + 12*a*b**5*d**(3/2)*x**(25/2)/25 + 2*b**6*d**(3/2)*x**(29/2)/29

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GIAC/XCAS [A]  time = 0.265365, size = 149, normalized size = 1.14 \[ \frac{2}{29} \, \sqrt{d x} b^{6} d x^{14} + \frac{12}{25} \, \sqrt{d x} a b^{5} d x^{12} + \frac{10}{7} \, \sqrt{d x} a^{2} b^{4} d x^{10} + \frac{40}{17} \, \sqrt{d x} a^{3} b^{3} d x^{8} + \frac{30}{13} \, \sqrt{d x} a^{4} b^{2} d x^{6} + \frac{4}{3} \, \sqrt{d x} a^{5} b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{6} d x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/29*sqrt(d*x)*b^6*d*x^14 + 12/25*sqrt(d*x)*a*b^5*d*x^12 + 10/7*sqrt(d*x)*a^2*b^
4*d*x^10 + 40/17*sqrt(d*x)*a^3*b^3*d*x^8 + 30/13*sqrt(d*x)*a^4*b^2*d*x^6 + 4/3*s
qrt(d*x)*a^5*b*d*x^4 + 2/5*sqrt(d*x)*a^6*d*x^2