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

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

[Out]

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

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Rubi [A]  time = 0.22648, 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 \[ \frac{2 b^5 (d x)^{23/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{19/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 d^9 \left (a+b x^2\right )}+\frac{4 a^2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^7 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.5441, size = 238, normalized size = 0.8 \[ \frac{16384 a^{5} \left (d x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{100947 d \left (a + b x^{2}\right )} + \frac{4096 a^{4} \left (d x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{33649 d} + \frac{512 a^{3} \left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{4807 d} + \frac{128 a^{2} \left (d x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{1311 d} + \frac{40 a \left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{437 d} + \frac{2 \left (d x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{23 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)*(d*x)**(1/2),x)

[Out]

16384*a**5*(d*x)**(3/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(100947*d*(a + b*x**
2)) + 4096*a**4*(d*x)**(3/2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(33649*d) + 512
*a**3*(d*x)**(3/2)*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(4807*d) + 1
28*a**2*(d*x)**(3/2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(1311*d) + 40*a*(d*x
)**(3/2)*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(437*d) + 2*(d*x)**
(3/2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(23*d)

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Mathematica [A]  time = 0.0467687, size = 88, normalized size = 0.3 \[ \frac{2 \sqrt{d x} \sqrt{\left (a+b x^2\right )^2} \left (33649 a^5 x+72105 a^4 b x^3+91770 a^3 b^2 x^5+67298 a^2 b^3 x^7+26565 a b^4 x^9+4389 b^5 x^{11}\right )}{100947 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d*x]*Sqrt[(a + b*x^2)^2]*(33649*a^5*x + 72105*a^4*b*x^3 + 91770*a^3*b^2*
x^5 + 67298*a^2*b^3*x^7 + 26565*a*b^4*x^9 + 4389*b^5*x^11))/(100947*(a + b*x^2))

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Maple [A]  time = 0.009, size = 83, normalized size = 0.3 \[{\frac{2\,x \left ( 4389\,{b}^{5}{x}^{10}+26565\,a{b}^{4}{x}^{8}+67298\,{a}^{2}{b}^{3}{x}^{6}+91770\,{a}^{3}{b}^{2}{x}^{4}+72105\,{a}^{4}b{x}^{2}+33649\,{a}^{5} \right ) }{100947\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}\sqrt{dx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)*(d*x)^(1/2),x)

[Out]

2/100947*x*(4389*b^5*x^10+26565*a*b^4*x^8+67298*a^2*b^3*x^6+91770*a^3*b^2*x^4+72
105*a^4*b*x^2+33649*a^5)*((b*x^2+a)^2)^(5/2)*(d*x)^(1/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.717625, size = 198, normalized size = 0.67 \[ \frac{2}{437} \,{\left (19 \, b^{5} \sqrt{d} x^{3} + 23 \, a b^{4} \sqrt{d} x\right )} x^{\frac{17}{2}} + \frac{8}{285} \,{\left (15 \, a b^{4} \sqrt{d} x^{3} + 19 \, a^{2} b^{3} \sqrt{d} x\right )} x^{\frac{13}{2}} + \frac{4}{55} \,{\left (11 \, a^{2} b^{3} \sqrt{d} x^{3} + 15 \, a^{3} b^{2} \sqrt{d} x\right )} x^{\frac{9}{2}} + \frac{8}{77} \,{\left (7 \, a^{3} b^{2} \sqrt{d} x^{3} + 11 \, a^{4} b \sqrt{d} x\right )} x^{\frac{5}{2}} + \frac{2}{21} \,{\left (3 \, a^{4} b \sqrt{d} x^{3} + 7 \, a^{5} \sqrt{d} x\right )} \sqrt{x} \]

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)*sqrt(d*x),x, algorithm="maxima")

[Out]

2/437*(19*b^5*sqrt(d)*x^3 + 23*a*b^4*sqrt(d)*x)*x^(17/2) + 8/285*(15*a*b^4*sqrt(
d)*x^3 + 19*a^2*b^3*sqrt(d)*x)*x^(13/2) + 4/55*(11*a^2*b^3*sqrt(d)*x^3 + 15*a^3*
b^2*sqrt(d)*x)*x^(9/2) + 8/77*(7*a^3*b^2*sqrt(d)*x^3 + 11*a^4*b*sqrt(d)*x)*x^(5/
2) + 2/21*(3*a^4*b*sqrt(d)*x^3 + 7*a^5*sqrt(d)*x)*sqrt(x)

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Fricas [A]  time = 0.269025, size = 84, normalized size = 0.28 \[ \frac{2}{100947} \,{\left (4389 \, b^{5} x^{11} + 26565 \, a b^{4} x^{9} + 67298 \, a^{2} b^{3} x^{7} + 91770 \, a^{3} b^{2} x^{5} + 72105 \, a^{4} b x^{3} + 33649 \, a^{5} x\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/100947*(4389*b^5*x^11 + 26565*a*b^4*x^9 + 67298*a^2*b^3*x^7 + 91770*a^3*b^2*x^
5 + 72105*a^4*b*x^3 + 33649*a^5*x)*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((b**2*x**4+2*a*b*x**2+a**2)**(5/2)*(d*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.267834, size = 194, normalized size = 0.65 \[ \frac{2 \,{\left (4389 \, \sqrt{d x} b^{5} d x^{11}{\rm sign}\left (b x^{2} + a\right ) + 26565 \, \sqrt{d x} a b^{4} d x^{9}{\rm sign}\left (b x^{2} + a\right ) + 67298 \, \sqrt{d x} a^{2} b^{3} d x^{7}{\rm sign}\left (b x^{2} + a\right ) + 91770 \, \sqrt{d x} a^{3} b^{2} d x^{5}{\rm sign}\left (b x^{2} + a\right ) + 72105 \, \sqrt{d x} a^{4} b d x^{3}{\rm sign}\left (b x^{2} + a\right ) + 33649 \, \sqrt{d x} a^{5} d x{\rm sign}\left (b x^{2} + a\right )\right )}}{100947 \, d} \]

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)*sqrt(d*x),x, algorithm="giac")

[Out]

2/100947*(4389*sqrt(d*x)*b^5*d*x^11*sign(b*x^2 + a) + 26565*sqrt(d*x)*a*b^4*d*x^
9*sign(b*x^2 + a) + 67298*sqrt(d*x)*a^2*b^3*d*x^7*sign(b*x^2 + a) + 91770*sqrt(d
*x)*a^3*b^2*d*x^5*sign(b*x^2 + a) + 72105*sqrt(d*x)*a^4*b*d*x^3*sign(b*x^2 + a)
+ 33649*sqrt(d*x)*a^5*d*x*sign(b*x^2 + a))/d