3.636 \(\int \frac{x^4 \left (a+b x^2\right )^2}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=194 \[ \frac{c^2 \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{9/2}}-\frac{c x \sqrt{c+d x^2} \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right )}{128 d^4}+\frac{x^3 \sqrt{c+d x^2} \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right )}{192 d^3}-\frac{b x^5 \sqrt{c+d x^2} (7 b c-16 a d)}{48 d^2}+\frac{b^2 x^7 \sqrt{c+d x^2}}{8 d} \]

[Out]

-(c*(48*a^2*d^2 + 5*b*c*(7*b*c - 16*a*d))*x*Sqrt[c + d*x^2])/(128*d^4) + ((48*a^
2*d^2 + 5*b*c*(7*b*c - 16*a*d))*x^3*Sqrt[c + d*x^2])/(192*d^3) - (b*(7*b*c - 16*
a*d)*x^5*Sqrt[c + d*x^2])/(48*d^2) + (b^2*x^7*Sqrt[c + d*x^2])/(8*d) + (c^2*(48*
a^2*d^2 + 5*b*c*(7*b*c - 16*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(
9/2))

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Rubi [A]  time = 0.461947, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{c^2 \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{9/2}}+\frac{x^3 \sqrt{c+d x^2} \left (48 a^2+\frac{5 b c (7 b c-16 a d)}{d^2}\right )}{192 d}-\frac{c x \sqrt{c+d x^2} \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right )}{128 d^4}-\frac{b x^5 \sqrt{c+d x^2} (7 b c-16 a d)}{48 d^2}+\frac{b^2 x^7 \sqrt{c+d x^2}}{8 d} \]

Antiderivative was successfully verified.

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

[Out]

-(c*(48*a^2*d^2 + 5*b*c*(7*b*c - 16*a*d))*x*Sqrt[c + d*x^2])/(128*d^4) + ((48*a^
2 + (5*b*c*(7*b*c - 16*a*d))/d^2)*x^3*Sqrt[c + d*x^2])/(192*d) - (b*(7*b*c - 16*
a*d)*x^5*Sqrt[c + d*x^2])/(48*d^2) + (b^2*x^7*Sqrt[c + d*x^2])/(8*d) + (c^2*(48*
a^2*d^2 + 5*b*c*(7*b*c - 16*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(
9/2))

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Rubi in Sympy [A]  time = 36.4558, size = 187, normalized size = 0.96 \[ \frac{b^{2} x^{7} \sqrt{c + d x^{2}}}{8 d} + \frac{b x^{5} \sqrt{c + d x^{2}} \left (16 a d - 7 b c\right )}{48 d^{2}} + \frac{c^{2} \left (48 a^{2} d^{2} - 5 b c \left (16 a d - 7 b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{128 d^{\frac{9}{2}}} - \frac{c x \sqrt{c + d x^{2}} \left (48 a^{2} d^{2} - 5 b c \left (16 a d - 7 b c\right )\right )}{128 d^{4}} + \frac{x^{3} \sqrt{c + d x^{2}} \left (48 a^{2} d^{2} - 5 b c \left (16 a d - 7 b c\right )\right )}{192 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**7*sqrt(c + d*x**2)/(8*d) + b*x**5*sqrt(c + d*x**2)*(16*a*d - 7*b*c)/(48*
d**2) + c**2*(48*a**2*d**2 - 5*b*c*(16*a*d - 7*b*c))*atanh(sqrt(d)*x/sqrt(c + d*
x**2))/(128*d**(9/2)) - c*x*sqrt(c + d*x**2)*(48*a**2*d**2 - 5*b*c*(16*a*d - 7*b
*c))/(128*d**4) + x**3*sqrt(c + d*x**2)*(48*a**2*d**2 - 5*b*c*(16*a*d - 7*b*c))/
(192*d**3)

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Mathematica [A]  time = 0.193349, size = 159, normalized size = 0.82 \[ \frac{3 c^2 \left (48 a^2 d^2-80 a b c d+35 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (48 a^2 d^2 \left (2 d x^2-3 c\right )+16 a b d \left (15 c^2-10 c d x^2+8 d^2 x^4\right )+b^2 \left (-105 c^3+70 c^2 d x^2-56 c d^2 x^4+48 d^3 x^6\right )\right )}{384 d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(48*a^2*d^2*(-3*c + 2*d*x^2) + 16*a*b*d*(15*c^2 - 10*
c*d*x^2 + 8*d^2*x^4) + b^2*(-105*c^3 + 70*c^2*d*x^2 - 56*c*d^2*x^4 + 48*d^3*x^6)
) + 3*c^2*(35*b^2*c^2 - 80*a*b*c*d + 48*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^
2]])/(384*d^(9/2))

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Maple [A]  time = 0.023, size = 265, normalized size = 1.4 \[{\frac{{a}^{2}{x}^{3}}{4\,d}\sqrt{d{x}^{2}+c}}-{\frac{3\,{a}^{2}cx}{8\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}{x}^{7}}{8\,d}\sqrt{d{x}^{2}+c}}-{\frac{7\,{b}^{2}c{x}^{5}}{48\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{35\,{b}^{2}{c}^{2}{x}^{3}}{192\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{35\,x{b}^{2}{c}^{3}}{128\,{d}^{4}}\sqrt{d{x}^{2}+c}}+{\frac{35\,{b}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{9}{2}}}}+{\frac{ab{x}^{5}}{3\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,abc{x}^{3}}{12\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{5\,ab{c}^{2}x}{8\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{5\,ab{c}^{3}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^2*x^3/d*(d*x^2+c)^(1/2)-3/8*a^2*c/d^2*x*(d*x^2+c)^(1/2)+3/8*a^2*c^2/d^(5/2
)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/8*b^2*x^7*(d*x^2+c)^(1/2)/d-7/48*b^2*c/d^2*x^5
*(d*x^2+c)^(1/2)+35/192*b^2*c^2/d^3*x^3*(d*x^2+c)^(1/2)-35/128*b^2*c^3/d^4*x*(d*
x^2+c)^(1/2)+35/128*b^2*c^4/d^(9/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/3*a*b*x^5/d*
(d*x^2+c)^(1/2)-5/12*a*b*c/d^2*x^3*(d*x^2+c)^(1/2)+5/8*a*b*c^2/d^3*x*(d*x^2+c)^(
1/2)-5/8*a*b*c^3/d^(7/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.375325, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, b^{2} d^{3} x^{7} - 8 \,{\left (7 \, b^{2} c d^{2} - 16 \, a b d^{3}\right )} x^{5} + 2 \,{\left (35 \, b^{2} c^{2} d - 80 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 80 \, a b c^{2} d + 48 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (35 \, b^{2} c^{4} - 80 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{768 \, d^{\frac{9}{2}}}, \frac{{\left (48 \, b^{2} d^{3} x^{7} - 8 \,{\left (7 \, b^{2} c d^{2} - 16 \, a b d^{3}\right )} x^{5} + 2 \,{\left (35 \, b^{2} c^{2} d - 80 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{3} - 80 \, a b c^{2} d + 48 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 3 \,{\left (35 \, b^{2} c^{4} - 80 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{384 \, \sqrt{-d} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(2*(48*b^2*d^3*x^7 - 8*(7*b^2*c*d^2 - 16*a*b*d^3)*x^5 + 2*(35*b^2*c^2*d -
80*a*b*c*d^2 + 48*a^2*d^3)*x^3 - 3*(35*b^2*c^3 - 80*a*b*c^2*d + 48*a^2*c*d^2)*x
)*sqrt(d*x^2 + c)*sqrt(d) + 3*(35*b^2*c^4 - 80*a*b*c^3*d + 48*a^2*c^2*d^2)*log(-
2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/d^(9/2), 1/384*((48*b^2*d^3*x^7
- 8*(7*b^2*c*d^2 - 16*a*b*d^3)*x^5 + 2*(35*b^2*c^2*d - 80*a*b*c*d^2 + 48*a^2*d^3
)*x^3 - 3*(35*b^2*c^3 - 80*a*b*c^2*d + 48*a^2*c*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d)
+ 3*(35*b^2*c^4 - 80*a*b*c^3*d + 48*a^2*c^2*d^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 +
c)))/(sqrt(-d)*d^4)]

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Sympy [A]  time = 63.9038, size = 422, normalized size = 2.18 \[ - \frac{3 a^{2} c^{\frac{3}{2}} x}{8 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} \sqrt{c} x^{3}}{8 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{5}{2}}} + \frac{a^{2} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b c^{\frac{5}{2}} x}{8 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b c^{\frac{3}{2}} x^{3}}{24 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b \sqrt{c} x^{5}}{12 d \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 a b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{7}{2}}} + \frac{a b x^{7}}{3 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{35 b^{2} c^{\frac{7}{2}} x}{128 d^{4} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{35 b^{2} c^{\frac{5}{2}} x^{3}}{384 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{7 b^{2} c^{\frac{3}{2}} x^{5}}{192 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} \sqrt{c} x^{7}}{48 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 b^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{9}{2}}} + \frac{b^{2} x^{9}}{8 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*c**(3/2)*x/(8*d**2*sqrt(1 + d*x**2/c)) - a**2*sqrt(c)*x**3/(8*d*sqrt(1 +
d*x**2/c)) + 3*a**2*c**2*asinh(sqrt(d)*x/sqrt(c))/(8*d**(5/2)) + a**2*x**5/(4*s
qrt(c)*sqrt(1 + d*x**2/c)) + 5*a*b*c**(5/2)*x/(8*d**3*sqrt(1 + d*x**2/c)) + 5*a*
b*c**(3/2)*x**3/(24*d**2*sqrt(1 + d*x**2/c)) - a*b*sqrt(c)*x**5/(12*d*sqrt(1 + d
*x**2/c)) - 5*a*b*c**3*asinh(sqrt(d)*x/sqrt(c))/(8*d**(7/2)) + a*b*x**7/(3*sqrt(
c)*sqrt(1 + d*x**2/c)) - 35*b**2*c**(7/2)*x/(128*d**4*sqrt(1 + d*x**2/c)) - 35*b
**2*c**(5/2)*x**3/(384*d**3*sqrt(1 + d*x**2/c)) + 7*b**2*c**(3/2)*x**5/(192*d**2
*sqrt(1 + d*x**2/c)) - b**2*sqrt(c)*x**7/(48*d*sqrt(1 + d*x**2/c)) + 35*b**2*c**
4*asinh(sqrt(d)*x/sqrt(c))/(128*d**(9/2)) + b**2*x**9/(8*sqrt(c)*sqrt(1 + d*x**2
/c))

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GIAC/XCAS [A]  time = 0.242375, size = 240, normalized size = 1.24 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (\frac{6 \, b^{2} x^{2}}{d} - \frac{7 \, b^{2} c d^{5} - 16 \, a b d^{6}}{d^{7}}\right )} x^{2} + \frac{35 \, b^{2} c^{2} d^{4} - 80 \, a b c d^{5} + 48 \, a^{2} d^{6}}{d^{7}}\right )} x^{2} - \frac{3 \,{\left (35 \, b^{2} c^{3} d^{3} - 80 \, a b c^{2} d^{4} + 48 \, a^{2} c d^{5}\right )}}{d^{7}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (35 \, b^{2} c^{4} - 80 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{128 \, d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(2*(4*(6*b^2*x^2/d - (7*b^2*c*d^5 - 16*a*b*d^6)/d^7)*x^2 + (35*b^2*c^2*d^4
- 80*a*b*c*d^5 + 48*a^2*d^6)/d^7)*x^2 - 3*(35*b^2*c^3*d^3 - 80*a*b*c^2*d^4 + 48
*a^2*c*d^5)/d^7)*sqrt(d*x^2 + c)*x - 1/128*(35*b^2*c^4 - 80*a*b*c^3*d + 48*a^2*c
^2*d^2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(9/2)