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

Optimal. Leaf size=297 \[ \frac{2 b^5 (d x)^{27/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{27 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac{4 a^3 b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )} \]

[Out]

(2*a^5*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d*(a + b*x^2)) + (10*a^4*
b*(d*x)^(11/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^3*(a + b*x^2)) + (4*a^3*b^
2*(d*x)^(15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d^5*(a + b*x^2)) + (20*a^2*b^
3*(d*x)^(19/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(19*d^7*(a + b*x^2)) + (10*a*b^4
*(d*x)^(23/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(23*d^9*(a + b*x^2)) + (2*b^5*(d*
x)^(27/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(27*d^11*(a + b*x^2))

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Rubi [A]  time = 0.238752, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b^5 (d x)^{27/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{27 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac{4 a^3 b^2 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^5*(d*x)^(7/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(7*d*(a + b*x^2)) + (10*a^4*
b*(d*x)^(11/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*d^3*(a + b*x^2)) + (4*a^3*b^
2*(d*x)^(15/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*d^5*(a + b*x^2)) + (20*a^2*b^
3*(d*x)^(19/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(19*d^7*(a + b*x^2)) + (10*a*b^4
*(d*x)^(23/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(23*d^9*(a + b*x^2)) + (2*b^5*(d*
x)^(27/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(27*d^11*(a + b*x^2))

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Rubi in Sympy [A]  time = 29.2781, size = 238, normalized size = 0.8 \[ \frac{16384 a^{5} \left (d x\right )^{\frac{7}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{908523 d \left (a + b x^{2}\right )} + \frac{4096 a^{4} \left (d x\right )^{\frac{7}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{129789 d} + \frac{512 a^{3} \left (d x\right )^{\frac{7}{2}} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{11799 d} + \frac{640 a^{2} \left (d x\right )^{\frac{7}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{11799 d} + \frac{40 a \left (d x\right )^{\frac{7}{2}} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{621 d} + \frac{2 \left (d x\right )^{\frac{7}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{27 d} \]

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)**(5/2),x)

[Out]

16384*a**5*(d*x)**(7/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(908523*d*(a + b*x**
2)) + 4096*a**4*(d*x)**(7/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(129789*d) + 51
2*a**3*(d*x)**(7/2)*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(11799*d) +
640*a**2*(d*x)**(7/2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(11799*d) + 40*a*(
d*x)**(7/2)*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(621*d) + 2*(d*x
)**(7/2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(27*d)

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Mathematica [A]  time = 0.0587774, size = 88, normalized size = 0.3 \[ \frac{2 x (d x)^{5/2} \sqrt{\left (a+b x^2\right )^2} \left (129789 a^5+412965 a^4 b x^2+605682 a^3 b^2 x^4+478170 a^2 b^3 x^6+197505 a b^4 x^8+33649 b^5 x^{10}\right )}{908523 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*x)^(5/2)*Sqrt[(a + b*x^2)^2]*(129789*a^5 + 412965*a^4*b*x^2 + 605682*a^3
*b^2*x^4 + 478170*a^2*b^3*x^6 + 197505*a*b^4*x^8 + 33649*b^5*x^10))/(908523*(a +
b*x^2))

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Maple [A]  time = 0.008, size = 83, normalized size = 0.3 \[{\frac{2\,x \left ( 33649\,{b}^{5}{x}^{10}+197505\,a{b}^{4}{x}^{8}+478170\,{a}^{2}{b}^{3}{x}^{6}+605682\,{a}^{3}{b}^{2}{x}^{4}+412965\,{a}^{4}b{x}^{2}+129789\,{a}^{5} \right ) }{908523\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( dx \right ) ^{{\frac{5}{2}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

2/908523*x*(33649*b^5*x^10+197505*a*b^4*x^8+478170*a^2*b^3*x^6+605682*a^3*b^2*x^
4+412965*a^4*b*x^2+129789*a^5)*(d*x)^(5/2)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.7288, size = 198, normalized size = 0.67 \[ \frac{2}{621} \,{\left (23 \, b^{5} d^{\frac{5}{2}} x^{3} + 27 \, a b^{4} d^{\frac{5}{2}} x\right )} x^{\frac{21}{2}} + \frac{8}{437} \,{\left (19 \, a b^{4} d^{\frac{5}{2}} x^{3} + 23 \, a^{2} b^{3} d^{\frac{5}{2}} x\right )} x^{\frac{17}{2}} + \frac{4}{95} \,{\left (15 \, a^{2} b^{3} d^{\frac{5}{2}} x^{3} + 19 \, a^{3} b^{2} d^{\frac{5}{2}} x\right )} x^{\frac{13}{2}} + \frac{8}{165} \,{\left (11 \, a^{3} b^{2} d^{\frac{5}{2}} x^{3} + 15 \, a^{4} b d^{\frac{5}{2}} x\right )} x^{\frac{9}{2}} + \frac{2}{77} \,{\left (7 \, a^{4} b d^{\frac{5}{2}} x^{3} + 11 \, a^{5} d^{\frac{5}{2}} x\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(5/2),x, algorithm="maxima")

[Out]

2/621*(23*b^5*d^(5/2)*x^3 + 27*a*b^4*d^(5/2)*x)*x^(21/2) + 8/437*(19*a*b^4*d^(5/
2)*x^3 + 23*a^2*b^3*d^(5/2)*x)*x^(17/2) + 4/95*(15*a^2*b^3*d^(5/2)*x^3 + 19*a^3*
b^2*d^(5/2)*x)*x^(13/2) + 8/165*(11*a^3*b^2*d^(5/2)*x^3 + 15*a^4*b*d^(5/2)*x)*x^
(9/2) + 2/77*(7*a^4*b*d^(5/2)*x^3 + 11*a^5*d^(5/2)*x)*x^(5/2)

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Fricas [A]  time = 0.274293, size = 111, normalized size = 0.37 \[ \frac{2}{908523} \,{\left (33649 \, b^{5} d^{2} x^{13} + 197505 \, a b^{4} d^{2} x^{11} + 478170 \, a^{2} b^{3} d^{2} x^{9} + 605682 \, a^{3} b^{2} d^{2} x^{7} + 412965 \, a^{4} b d^{2} x^{5} + 129789 \, a^{5} d^{2} x^{3}\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(5/2),x, algorithm="fricas")

[Out]

2/908523*(33649*b^5*d^2*x^13 + 197505*a*b^4*d^2*x^11 + 478170*a^2*b^3*d^2*x^9 +
605682*a^3*b^2*d^2*x^7 + 412965*a^4*b*d^2*x^5 + 129789*a^5*d^2*x^3)*sqrt(d*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.268335, size = 207, normalized size = 0.7 \[ \frac{2}{27} \, \sqrt{d x} b^{5} d^{2} x^{13}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{23} \, \sqrt{d x} a b^{4} d^{2} x^{11}{\rm sign}\left (b x^{2} + a\right ) + \frac{20}{19} \, \sqrt{d x} a^{2} b^{3} d^{2} x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{4}{3} \, \sqrt{d x} a^{3} b^{2} d^{2} x^{7}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{11} \, \sqrt{d x} a^{4} b d^{2} x^{5}{\rm sign}\left (b x^{2} + a\right ) + \frac{2}{7} \, \sqrt{d x} a^{5} d^{2} x^{3}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(5/2),x, algorithm="giac")

[Out]

2/27*sqrt(d*x)*b^5*d^2*x^13*sign(b*x^2 + a) + 10/23*sqrt(d*x)*a*b^4*d^2*x^11*sig
n(b*x^2 + a) + 20/19*sqrt(d*x)*a^2*b^3*d^2*x^9*sign(b*x^2 + a) + 4/3*sqrt(d*x)*a
^3*b^2*d^2*x^7*sign(b*x^2 + a) + 10/11*sqrt(d*x)*a^4*b*d^2*x^5*sign(b*x^2 + a) +
2/7*sqrt(d*x)*a^5*d^2*x^3*sign(b*x^2 + a)