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

Optimal. Leaf size=89 \[ -\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}+\frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]

[Out]

(-2*a^4)/(d*Sqrt[d*x]) + (8*a^3*b*(d*x)^(3/2))/(3*d^3) + (12*a^2*b^2*(d*x)^(7/2)
)/(7*d^5) + (8*a*b^3*(d*x)^(11/2))/(11*d^7) + (2*b^4*(d*x)^(15/2))/(15*d^9)

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Rubi [A]  time = 0.104692, antiderivative size = 89, 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 \[ -\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}+\frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^4)/(d*Sqrt[d*x]) + (8*a^3*b*(d*x)^(3/2))/(3*d^3) + (12*a^2*b^2*(d*x)^(7/2)
)/(7*d^5) + (8*a*b^3*(d*x)^(11/2))/(11*d^7) + (2*b^4*(d*x)^(15/2))/(15*d^9)

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Rubi in Sympy [A]  time = 25.8596, size = 87, normalized size = 0.98 \[ - \frac{2 a^{4}}{d \sqrt{d x}} + \frac{8 a^{3} b \left (d x\right )^{\frac{3}{2}}}{3 d^{3}} + \frac{12 a^{2} b^{2} \left (d x\right )^{\frac{7}{2}}}{7 d^{5}} + \frac{8 a b^{3} \left (d x\right )^{\frac{11}{2}}}{11 d^{7}} + \frac{2 b^{4} \left (d x\right )^{\frac{15}{2}}}{15 d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**4/(d*sqrt(d*x)) + 8*a**3*b*(d*x)**(3/2)/(3*d**3) + 12*a**2*b**2*(d*x)**(7/
2)/(7*d**5) + 8*a*b**3*(d*x)**(11/2)/(11*d**7) + 2*b**4*(d*x)**(15/2)/(15*d**9)

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Mathematica [A]  time = 0.0215086, size = 55, normalized size = 0.62 \[ \frac{2 x \left (-1155 a^4+1540 a^3 b x^2+990 a^2 b^2 x^4+420 a b^3 x^6+77 b^4 x^8\right )}{1155 (d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-1155*a^4 + 1540*a^3*b*x^2 + 990*a^2*b^2*x^4 + 420*a*b^3*x^6 + 77*b^4*x^8)
)/(1155*(d*x)^(3/2))

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Maple [A]  time = 0.01, size = 52, normalized size = 0.6 \[ -{\frac{ \left ( -154\,{b}^{4}{x}^{8}-840\,a{b}^{3}{x}^{6}-1980\,{a}^{2}{b}^{2}{x}^{4}-3080\,{a}^{3}b{x}^{2}+2310\,{a}^{4} \right ) x}{1155} \left ( dx \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(-77*b^4*x^8-420*a*b^3*x^6-990*a^2*b^2*x^4-1540*a^3*b*x^2+1155*a^4)*x/(d
*x)^(3/2)

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Maxima [A]  time = 0.693294, size = 103, normalized size = 1.16 \[ -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \left (d x\right )^{\frac{15}{2}} b^{4} + 420 \, \left (d x\right )^{\frac{11}{2}} a b^{3} d^{2} + 990 \, \left (d x\right )^{\frac{7}{2}} a^{2} b^{2} d^{4} + 1540 \, \left (d x\right )^{\frac{3}{2}} a^{3} b d^{6}}{d^{8}}\right )}}{1155 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(1155*a^4/sqrt(d*x) - (77*(d*x)^(15/2)*b^4 + 420*(d*x)^(11/2)*a*b^3*d^2
+ 990*(d*x)^(7/2)*a^2*b^2*d^4 + 1540*(d*x)^(3/2)*a^3*b*d^6)/d^8)/d

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Fricas [A]  time = 0.260165, size = 72, normalized size = 0.81 \[ \frac{2 \,{\left (77 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 990 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} - 1155 \, a^{4}\right )}}{1155 \, \sqrt{d x} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*(77*b^4*x^8 + 420*a*b^3*x^6 + 990*a^2*b^2*x^4 + 1540*a^3*b*x^2 - 1155*a^4
)/(sqrt(d*x)*d)

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Sympy [A]  time = 4.82865, size = 88, normalized size = 0.99 \[ - \frac{2 a^{4}}{d^{\frac{3}{2}} \sqrt{x}} + \frac{8 a^{3} b x^{\frac{3}{2}}}{3 d^{\frac{3}{2}}} + \frac{12 a^{2} b^{2} x^{\frac{7}{2}}}{7 d^{\frac{3}{2}}} + \frac{8 a b^{3} x^{\frac{11}{2}}}{11 d^{\frac{3}{2}}} + \frac{2 b^{4} x^{\frac{15}{2}}}{15 d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**4/(d**(3/2)*sqrt(x)) + 8*a**3*b*x**(3/2)/(3*d**(3/2)) + 12*a**2*b**2*x**(7
/2)/(7*d**(3/2)) + 8*a*b**3*x**(11/2)/(11*d**(3/2)) + 2*b**4*x**(15/2)/(15*d**(3
/2))

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GIAC/XCAS [A]  time = 0.264769, size = 120, normalized size = 1.35 \[ -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \sqrt{d x} b^{4} d^{119} x^{7} + 420 \, \sqrt{d x} a b^{3} d^{119} x^{5} + 990 \, \sqrt{d x} a^{2} b^{2} d^{119} x^{3} + 1540 \, \sqrt{d x} a^{3} b d^{119} x}{d^{120}}\right )}}{1155 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(1155*a^4/sqrt(d*x) - (77*sqrt(d*x)*b^4*d^119*x^7 + 420*sqrt(d*x)*a*b^3*
d^119*x^5 + 990*sqrt(d*x)*a^2*b^2*d^119*x^3 + 1540*sqrt(d*x)*a^3*b*d^119*x)/d^12
0)/d