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

Optimal. Leaf size=131 \[ \frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}} \]

[Out]

(2*a^6*(d*x)^(3/2))/(3*d) + (12*a^5*b*(d*x)^(7/2))/(7*d^3) + (30*a^4*b^2*(d*x)^(
11/2))/(11*d^5) + (8*a^3*b^3*(d*x)^(15/2))/(3*d^7) + (30*a^2*b^4*(d*x)^(19/2))/(
19*d^9) + (12*a*b^5*(d*x)^(23/2))/(23*d^11) + (2*b^6*(d*x)^(27/2))/(27*d^13)

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Rubi [A]  time = 0.151779, 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 \[ \frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^6*(d*x)^(3/2))/(3*d) + (12*a^5*b*(d*x)^(7/2))/(7*d^3) + (30*a^4*b^2*(d*x)^(
11/2))/(11*d^5) + (8*a^3*b^3*(d*x)^(15/2))/(3*d^7) + (30*a^2*b^4*(d*x)^(19/2))/(
19*d^9) + (12*a*b^5*(d*x)^(23/2))/(23*d^11) + (2*b^6*(d*x)^(27/2))/(27*d^13)

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Rubi in Sympy [A]  time = 36.0538, size = 129, normalized size = 0.98 \[ \frac{2 a^{6} \left (d x\right )^{\frac{3}{2}}}{3 d} + \frac{12 a^{5} b \left (d x\right )^{\frac{7}{2}}}{7 d^{3}} + \frac{30 a^{4} b^{2} \left (d x\right )^{\frac{11}{2}}}{11 d^{5}} + \frac{8 a^{3} b^{3} \left (d x\right )^{\frac{15}{2}}}{3 d^{7}} + \frac{30 a^{2} b^{4} \left (d x\right )^{\frac{19}{2}}}{19 d^{9}} + \frac{12 a b^{5} \left (d x\right )^{\frac{23}{2}}}{23 d^{11}} + \frac{2 b^{6} \left (d x\right )^{\frac{27}{2}}}{27 d^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**3*(d*x)**(1/2),x)

[Out]

2*a**6*(d*x)**(3/2)/(3*d) + 12*a**5*b*(d*x)**(7/2)/(7*d**3) + 30*a**4*b**2*(d*x)
**(11/2)/(11*d**5) + 8*a**3*b**3*(d*x)**(15/2)/(3*d**7) + 30*a**2*b**4*(d*x)**(1
9/2)/(19*d**9) + 12*a*b**5*(d*x)**(23/2)/(23*d**11) + 2*b**6*(d*x)**(27/2)/(27*d
**13)

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Mathematica [A]  time = 0.0235069, size = 77, normalized size = 0.59 \[ \frac{2 x \sqrt{d x} \left (302841 a^6+778734 a^5 b x^2+1238895 a^4 b^2 x^4+1211364 a^3 b^3 x^6+717255 a^2 b^4 x^8+237006 a b^5 x^{10}+33649 b^6 x^{12}\right )}{908523} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[d*x]*(302841*a^6 + 778734*a^5*b*x^2 + 1238895*a^4*b^2*x^4 + 1211364*a^
3*b^3*x^6 + 717255*a^2*b^4*x^8 + 237006*a*b^5*x^10 + 33649*b^6*x^12))/908523

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Maple [A]  time = 0.011, size = 74, normalized size = 0.6 \[{\frac{2\,x \left ( 33649\,{b}^{6}{x}^{12}+237006\,a{b}^{5}{x}^{10}+717255\,{a}^{2}{b}^{4}{x}^{8}+1211364\,{a}^{3}{b}^{3}{x}^{6}+1238895\,{a}^{4}{b}^{2}{x}^{4}+778734\,{a}^{5}b{x}^{2}+302841\,{a}^{6} \right ) }{908523}\sqrt{dx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^3*(d*x)^(1/2),x)

[Out]

2/908523*x*(33649*b^6*x^12+237006*a*b^5*x^10+717255*a^2*b^4*x^8+1211364*a^3*b^3*
x^6+1238895*a^4*b^2*x^4+778734*a^5*b*x^2+302841*a^6)*(d*x)^(1/2)

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Maxima [A]  time = 0.703632, size = 142, normalized size = 1.08 \[ \frac{2 \,{\left (33649 \, \left (d x\right )^{\frac{27}{2}} b^{6} + 237006 \, \left (d x\right )^{\frac{23}{2}} a b^{5} d^{2} + 717255 \, \left (d x\right )^{\frac{19}{2}} a^{2} b^{4} d^{4} + 1211364 \, \left (d x\right )^{\frac{15}{2}} a^{3} b^{3} d^{6} + 1238895 \, \left (d x\right )^{\frac{11}{2}} a^{4} b^{2} d^{8} + 778734 \, \left (d x\right )^{\frac{7}{2}} a^{5} b d^{10} + 302841 \, \left (d x\right )^{\frac{3}{2}} a^{6} d^{12}\right )}}{908523 \, d^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/908523*(33649*(d*x)^(27/2)*b^6 + 237006*(d*x)^(23/2)*a*b^5*d^2 + 717255*(d*x)^
(19/2)*a^2*b^4*d^4 + 1211364*(d*x)^(15/2)*a^3*b^3*d^6 + 1238895*(d*x)^(11/2)*a^4
*b^2*d^8 + 778734*(d*x)^(7/2)*a^5*b*d^10 + 302841*(d*x)^(3/2)*a^6*d^12)/d^13

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Fricas [A]  time = 0.257206, size = 99, normalized size = 0.76 \[ \frac{2}{908523} \,{\left (33649 \, b^{6} x^{13} + 237006 \, a b^{5} x^{11} + 717255 \, a^{2} b^{4} x^{9} + 1211364 \, a^{3} b^{3} x^{7} + 1238895 \, a^{4} b^{2} x^{5} + 778734 \, a^{5} b x^{3} + 302841 \, a^{6} x\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/908523*(33649*b^6*x^13 + 237006*a*b^5*x^11 + 717255*a^2*b^4*x^9 + 1211364*a^3*
b^3*x^7 + 1238895*a^4*b^2*x^5 + 778734*a^5*b*x^3 + 302841*a^6*x)*sqrt(d*x)

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Sympy [A]  time = 9.01914, size = 131, normalized size = 1. \[ \frac{2 a^{6} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{12 a^{5} b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{30 a^{4} b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} + \frac{8 a^{3} b^{3} \sqrt{d} x^{\frac{15}{2}}}{3} + \frac{30 a^{2} b^{4} \sqrt{d} x^{\frac{19}{2}}}{19} + \frac{12 a b^{5} \sqrt{d} x^{\frac{23}{2}}}{23} + \frac{2 b^{6} \sqrt{d} x^{\frac{27}{2}}}{27} \]

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)**(1/2),x)

[Out]

2*a**6*sqrt(d)*x**(3/2)/3 + 12*a**5*b*sqrt(d)*x**(7/2)/7 + 30*a**4*b**2*sqrt(d)*
x**(11/2)/11 + 8*a**3*b**3*sqrt(d)*x**(15/2)/3 + 30*a**2*b**4*sqrt(d)*x**(19/2)/
19 + 12*a*b**5*sqrt(d)*x**(23/2)/23 + 2*b**6*sqrt(d)*x**(27/2)/27

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GIAC/XCAS [A]  time = 0.264919, size = 153, normalized size = 1.17 \[ \frac{2 \,{\left (33649 \, \sqrt{d x} b^{6} d x^{13} + 237006 \, \sqrt{d x} a b^{5} d x^{11} + 717255 \, \sqrt{d x} a^{2} b^{4} d x^{9} + 1211364 \, \sqrt{d x} a^{3} b^{3} d x^{7} + 1238895 \, \sqrt{d x} a^{4} b^{2} d x^{5} + 778734 \, \sqrt{d x} a^{5} b d x^{3} + 302841 \, \sqrt{d x} a^{6} d x\right )}}{908523 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/908523*(33649*sqrt(d*x)*b^6*d*x^13 + 237006*sqrt(d*x)*a*b^5*d*x^11 + 717255*sq
rt(d*x)*a^2*b^4*d*x^9 + 1211364*sqrt(d*x)*a^3*b^3*d*x^7 + 1238895*sqrt(d*x)*a^4*
b^2*d*x^5 + 778734*sqrt(d*x)*a^5*b*d*x^3 + 302841*sqrt(d*x)*a^6*d*x)/d