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

Optimal. Leaf size=297 \[ \frac{2 b^5 (d x)^{25/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{25 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )} \]

[Out]

(2*a^5*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) + (10*a^4*
b*(d*x)^(9/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^3*(a + b*x^2)) + (20*a^3*b^2
*(d*x)^(13/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*d^5*(a + b*x^2)) + (20*a^2*b^
3*(d*x)^(17/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*d^7*(a + b*x^2)) + (10*a*b^4
*(d*x)^(21/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(21*d^9*(a + b*x^2)) + (2*b^5*(d*
x)^(25/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(25*d^11*(a + b*x^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.226831, 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 \[ \frac{2 b^5 (d x)^{25/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{25 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^5*(d*x)^(5/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*d*(a + b*x^2)) + (10*a^4*
b*(d*x)^(9/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(9*d^3*(a + b*x^2)) + (20*a^3*b^2
*(d*x)^(13/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*d^5*(a + b*x^2)) + (20*a^2*b^
3*(d*x)^(17/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(17*d^7*(a + b*x^2)) + (10*a*b^4
*(d*x)^(21/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(21*d^9*(a + b*x^2)) + (2*b^5*(d*
x)^(25/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(25*d^11*(a + b*x^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.1006, size = 238, normalized size = 0.8 \[ \frac{16384 a^{5} \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{348075 d \left (a + b x^{2}\right )} + \frac{4096 a^{4} \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{69615 d} + \frac{512 a^{3} \left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{7735 d} + \frac{128 a^{2} \left (d x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{1785 d} + \frac{8 a \left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{105 d} + \frac{2 \left (d x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{25 d} \]

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)**(5/2),x)

[Out]

16384*a**5*(d*x)**(5/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(348075*d*(a + b*x**
2)) + 4096*a**4*(d*x)**(5/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(69615*d) + 512
*a**3*(d*x)**(5/2)*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(7735*d) + 1
28*a**2*(d*x)**(5/2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(1785*d) + 8*a*(d*x)
**(5/2)*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(105*d) + 2*(d*x)**(
5/2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(25*d)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0463342, size = 88, normalized size = 0.3 \[ \frac{2 x (d x)^{3/2} \sqrt{\left (a+b x^2\right )^2} \left (69615 a^5+193375 a^4 b x^2+267750 a^3 b^2 x^4+204750 a^2 b^3 x^6+82875 a b^4 x^8+13923 b^5 x^{10}\right )}{348075 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*x)^(3/2)*Sqrt[(a + b*x^2)^2]*(69615*a^5 + 193375*a^4*b*x^2 + 267750*a^3*
b^2*x^4 + 204750*a^2*b^3*x^6 + 82875*a*b^4*x^8 + 13923*b^5*x^10))/(348075*(a + b
*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 83, normalized size = 0.3 \[{\frac{2\,x \left ( 13923\,{b}^{5}{x}^{10}+82875\,a{b}^{4}{x}^{8}+204750\,{a}^{2}{b}^{3}{x}^{6}+267750\,{a}^{3}{b}^{2}{x}^{4}+193375\,{a}^{4}b{x}^{2}+69615\,{a}^{5} \right ) }{348075\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( dx \right ) ^{{\frac{3}{2}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/348075*x*(13923*b^5*x^10+82875*a*b^4*x^8+204750*a^2*b^3*x^6+267750*a^3*b^2*x^4
+193375*a^4*b*x^2+69615*a^5)*(d*x)^(3/2)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

_______________________________________________________________________________________

Maxima [A]  time = 0.721859, size = 198, normalized size = 0.67 \[ \frac{2}{525} \,{\left (21 \, b^{5} d^{\frac{3}{2}} x^{3} + 25 \, a b^{4} d^{\frac{3}{2}} x\right )} x^{\frac{19}{2}} + \frac{8}{357} \,{\left (17 \, a b^{4} d^{\frac{3}{2}} x^{3} + 21 \, a^{2} b^{3} d^{\frac{3}{2}} x\right )} x^{\frac{15}{2}} + \frac{12}{221} \,{\left (13 \, a^{2} b^{3} d^{\frac{3}{2}} x^{3} + 17 \, a^{3} b^{2} d^{\frac{3}{2}} x\right )} x^{\frac{11}{2}} + \frac{8}{117} \,{\left (9 \, a^{3} b^{2} d^{\frac{3}{2}} x^{3} + 13 \, a^{4} b d^{\frac{3}{2}} x\right )} x^{\frac{7}{2}} + \frac{2}{45} \,{\left (5 \, a^{4} b d^{\frac{3}{2}} x^{3} + 9 \, a^{5} d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/525*(21*b^5*d^(3/2)*x^3 + 25*a*b^4*d^(3/2)*x)*x^(19/2) + 8/357*(17*a*b^4*d^(3/
2)*x^3 + 21*a^2*b^3*d^(3/2)*x)*x^(15/2) + 12/221*(13*a^2*b^3*d^(3/2)*x^3 + 17*a^
3*b^2*d^(3/2)*x)*x^(11/2) + 8/117*(9*a^3*b^2*d^(3/2)*x^3 + 13*a^4*b*d^(3/2)*x)*x
^(7/2) + 2/45*(5*a^4*b*d^(3/2)*x^3 + 9*a^5*d^(3/2)*x)*x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.272575, size = 95, normalized size = 0.32 \[ \frac{2}{348075} \,{\left (13923 \, b^{5} d x^{12} + 82875 \, a b^{4} d x^{10} + 204750 \, a^{2} b^{3} d x^{8} + 267750 \, a^{3} b^{2} d x^{6} + 193375 \, a^{4} b d x^{4} + 69615 \, a^{5} d x^{2}\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(3/2),x, algorithm="fricas")

[Out]

2/348075*(13923*b^5*d*x^12 + 82875*a*b^4*d*x^10 + 204750*a^2*b^3*d*x^8 + 267750*
a^3*b^2*d*x^6 + 193375*a^4*b*d*x^4 + 69615*a^5*d*x^2)*sqrt(d*x)

_______________________________________________________________________________________

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)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269821, size = 190, normalized size = 0.64 \[ \frac{2}{25} \, \sqrt{d x} b^{5} d x^{12}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{21} \, \sqrt{d x} a b^{4} d x^{10}{\rm sign}\left (b x^{2} + a\right ) + \frac{20}{17} \, \sqrt{d x} a^{2} b^{3} d x^{8}{\rm sign}\left (b x^{2} + a\right ) + \frac{20}{13} \, \sqrt{d x} a^{3} b^{2} d x^{6}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{9} \, \sqrt{d x} a^{4} b d x^{4}{\rm sign}\left (b x^{2} + a\right ) + \frac{2}{5} \, \sqrt{d x} a^{5} d x^{2}{\rm sign}\left (b x^{2} + a\right ) \]

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)^(3/2),x, algorithm="giac")

[Out]

2/25*sqrt(d*x)*b^5*d*x^12*sign(b*x^2 + a) + 10/21*sqrt(d*x)*a*b^4*d*x^10*sign(b*
x^2 + a) + 20/17*sqrt(d*x)*a^2*b^3*d*x^8*sign(b*x^2 + a) + 20/13*sqrt(d*x)*a^3*b
^2*d*x^6*sign(b*x^2 + a) + 10/9*sqrt(d*x)*a^4*b*d*x^4*sign(b*x^2 + a) + 2/5*sqrt
(d*x)*a^5*d*x^2*sign(b*x^2 + a)