3.679 \(\int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx\)

Optimal. Leaf size=129 \[ \frac{2 a^6 (d x)^{7/2}}{7 d}+\frac{12 a^5 b (d x)^{11/2}}{11 d^3}+\frac{2 a^4 b^2 (d x)^{15/2}}{d^5}+\frac{40 a^3 b^3 (d x)^{19/2}}{19 d^7}+\frac{30 a^2 b^4 (d x)^{23/2}}{23 d^9}+\frac{4 a b^5 (d x)^{27/2}}{9 d^{11}}+\frac{2 b^6 (d x)^{31/2}}{31 d^{13}} \]

[Out]

(2*a^6*(d*x)^(7/2))/(7*d) + (12*a^5*b*(d*x)^(11/2))/(11*d^3) + (2*a^4*b^2*(d*x)^
(15/2))/d^5 + (40*a^3*b^3*(d*x)^(19/2))/(19*d^7) + (30*a^2*b^4*(d*x)^(23/2))/(23
*d^9) + (4*a*b^5*(d*x)^(27/2))/(9*d^11) + (2*b^6*(d*x)^(31/2))/(31*d^13)

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Rubi [A]  time = 0.17419, antiderivative size = 129, 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 \[ \frac{2 a^6 (d x)^{7/2}}{7 d}+\frac{12 a^5 b (d x)^{11/2}}{11 d^3}+\frac{2 a^4 b^2 (d x)^{15/2}}{d^5}+\frac{40 a^3 b^3 (d x)^{19/2}}{19 d^7}+\frac{30 a^2 b^4 (d x)^{23/2}}{23 d^9}+\frac{4 a b^5 (d x)^{27/2}}{9 d^{11}}+\frac{2 b^6 (d x)^{31/2}}{31 d^{13}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*a^6*(d*x)^(7/2))/(7*d) + (12*a^5*b*(d*x)^(11/2))/(11*d^3) + (2*a^4*b^2*(d*x)^
(15/2))/d^5 + (40*a^3*b^3*(d*x)^(19/2))/(19*d^7) + (30*a^2*b^4*(d*x)^(23/2))/(23
*d^9) + (4*a*b^5*(d*x)^(27/2))/(9*d^11) + (2*b^6*(d*x)^(31/2))/(31*d^13)

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Rubi in Sympy [A]  time = 35.9814, size = 128, normalized size = 0.99 \[ \frac{2 a^{6} \left (d x\right )^{\frac{7}{2}}}{7 d} + \frac{12 a^{5} b \left (d x\right )^{\frac{11}{2}}}{11 d^{3}} + \frac{2 a^{4} b^{2} \left (d x\right )^{\frac{15}{2}}}{d^{5}} + \frac{40 a^{3} b^{3} \left (d x\right )^{\frac{19}{2}}}{19 d^{7}} + \frac{30 a^{2} b^{4} \left (d x\right )^{\frac{23}{2}}}{23 d^{9}} + \frac{4 a b^{5} \left (d x\right )^{\frac{27}{2}}}{9 d^{11}} + \frac{2 b^{6} \left (d x\right )^{\frac{31}{2}}}{31 d^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(5/2)*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

2*a**6*(d*x)**(7/2)/(7*d) + 12*a**5*b*(d*x)**(11/2)/(11*d**3) + 2*a**4*b**2*(d*x
)**(15/2)/d**5 + 40*a**3*b**3*(d*x)**(19/2)/(19*d**7) + 30*a**2*b**4*(d*x)**(23/
2)/(23*d**9) + 4*a*b**5*(d*x)**(27/2)/(9*d**11) + 2*b**6*(d*x)**(31/2)/(31*d**13
)

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Mathematica [A]  time = 0.034641, size = 77, normalized size = 0.6 \[ \frac{2 x (d x)^{5/2} \left (1341153 a^6+5120766 a^5 b x^2+9388071 a^4 b^2 x^4+9882180 a^3 b^3 x^6+6122655 a^2 b^4 x^8+2086238 a b^5 x^{10}+302841 b^6 x^{12}\right )}{9388071} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*x*(d*x)^(5/2)*(1341153*a^6 + 5120766*a^5*b*x^2 + 9388071*a^4*b^2*x^4 + 988218
0*a^3*b^3*x^6 + 6122655*a^2*b^4*x^8 + 2086238*a*b^5*x^10 + 302841*b^6*x^12))/938
8071

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Maple [A]  time = 0.011, size = 74, normalized size = 0.6 \[{\frac{2\,x \left ( 302841\,{b}^{6}{x}^{12}+2086238\,a{b}^{5}{x}^{10}+6122655\,{a}^{2}{b}^{4}{x}^{8}+9882180\,{a}^{3}{b}^{3}{x}^{6}+9388071\,{a}^{4}{b}^{2}{x}^{4}+5120766\,{a}^{5}b{x}^{2}+1341153\,{a}^{6} \right ) }{9388071} \left ( dx \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(5/2)*(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

2/9388071*x*(302841*b^6*x^12+2086238*a*b^5*x^10+6122655*a^2*b^4*x^8+9882180*a^3*
b^3*x^6+9388071*a^4*b^2*x^4+5120766*a^5*b*x^2+1341153*a^6)*(d*x)^(5/2)

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Maxima [A]  time = 0.688189, size = 142, normalized size = 1.1 \[ \frac{2 \,{\left (302841 \, \left (d x\right )^{\frac{31}{2}} b^{6} + 2086238 \, \left (d x\right )^{\frac{27}{2}} a b^{5} d^{2} + 6122655 \, \left (d x\right )^{\frac{23}{2}} a^{2} b^{4} d^{4} + 9882180 \, \left (d x\right )^{\frac{19}{2}} a^{3} b^{3} d^{6} + 9388071 \, \left (d x\right )^{\frac{15}{2}} a^{4} b^{2} d^{8} + 5120766 \, \left (d x\right )^{\frac{11}{2}} a^{5} b d^{10} + 1341153 \, \left (d x\right )^{\frac{7}{2}} a^{6} d^{12}\right )}}{9388071 \, d^{13}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/9388071*(302841*(d*x)^(31/2)*b^6 + 2086238*(d*x)^(27/2)*a*b^5*d^2 + 6122655*(d
*x)^(23/2)*a^2*b^4*d^4 + 9882180*(d*x)^(19/2)*a^3*b^3*d^6 + 9388071*(d*x)^(15/2)
*a^4*b^2*d^8 + 5120766*(d*x)^(11/2)*a^5*b*d^10 + 1341153*(d*x)^(7/2)*a^6*d^12)/d
^13

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Fricas [A]  time = 0.256801, size = 130, normalized size = 1.01 \[ \frac{2}{9388071} \,{\left (302841 \, b^{6} d^{2} x^{15} + 2086238 \, a b^{5} d^{2} x^{13} + 6122655 \, a^{2} b^{4} d^{2} x^{11} + 9882180 \, a^{3} b^{3} d^{2} x^{9} + 9388071 \, a^{4} b^{2} d^{2} x^{7} + 5120766 \, a^{5} b d^{2} x^{5} + 1341153 \, a^{6} d^{2} x^{3}\right )} \sqrt{d x} \]

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)^(5/2),x, algorithm="fricas")

[Out]

2/9388071*(302841*b^6*d^2*x^15 + 2086238*a*b^5*d^2*x^13 + 6122655*a^2*b^4*d^2*x^
11 + 9882180*a^3*b^3*d^2*x^9 + 9388071*a^4*b^2*d^2*x^7 + 5120766*a^5*b*d^2*x^5 +
1341153*a^6*d^2*x^3)*sqrt(d*x)

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Sympy [A]  time = 33.7286, size = 129, normalized size = 1. \[ \frac{2 a^{6} d^{\frac{5}{2}} x^{\frac{7}{2}}}{7} + \frac{12 a^{5} b d^{\frac{5}{2}} x^{\frac{11}{2}}}{11} + 2 a^{4} b^{2} d^{\frac{5}{2}} x^{\frac{15}{2}} + \frac{40 a^{3} b^{3} d^{\frac{5}{2}} x^{\frac{19}{2}}}{19} + \frac{30 a^{2} b^{4} d^{\frac{5}{2}} x^{\frac{23}{2}}}{23} + \frac{4 a b^{5} d^{\frac{5}{2}} x^{\frac{27}{2}}}{9} + \frac{2 b^{6} d^{\frac{5}{2}} x^{\frac{31}{2}}}{31} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**(5/2)*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

2*a**6*d**(5/2)*x**(7/2)/7 + 12*a**5*b*d**(5/2)*x**(11/2)/11 + 2*a**4*b**2*d**(5
/2)*x**(15/2) + 40*a**3*b**3*d**(5/2)*x**(19/2)/19 + 30*a**2*b**4*d**(5/2)*x**(2
3/2)/23 + 4*a*b**5*d**(5/2)*x**(27/2)/9 + 2*b**6*d**(5/2)*x**(31/2)/31

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GIAC/XCAS [A]  time = 0.265429, size = 167, normalized size = 1.29 \[ \frac{2}{31} \, \sqrt{d x} b^{6} d^{2} x^{15} + \frac{4}{9} \, \sqrt{d x} a b^{5} d^{2} x^{13} + \frac{30}{23} \, \sqrt{d x} a^{2} b^{4} d^{2} x^{11} + \frac{40}{19} \, \sqrt{d x} a^{3} b^{3} d^{2} x^{9} + 2 \, \sqrt{d x} a^{4} b^{2} d^{2} x^{7} + \frac{12}{11} \, \sqrt{d x} a^{5} b d^{2} x^{5} + \frac{2}{7} \, \sqrt{d x} a^{6} d^{2} x^{3} \]

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)^(5/2),x, algorithm="giac")

[Out]

2/31*sqrt(d*x)*b^6*d^2*x^15 + 4/9*sqrt(d*x)*a*b^5*d^2*x^13 + 30/23*sqrt(d*x)*a^2
*b^4*d^2*x^11 + 40/19*sqrt(d*x)*a^3*b^3*d^2*x^9 + 2*sqrt(d*x)*a^4*b^2*d^2*x^7 +
12/11*sqrt(d*x)*a^5*b*d^2*x^5 + 2/7*sqrt(d*x)*a^6*d^2*x^3