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

Optimal. Leaf size=191 \[ -\frac{c^2 \left (16 a^2 d^2+b c (5 b c-16 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{7/2}}+\frac{x^3 \sqrt{c+d x^2} \left (16 a^2 d^2+b c (5 b c-16 a d)\right )}{64 d^2}+\frac{c x \sqrt{c+d x^2} \left (16 a^2 d^2+b c (5 b c-16 a d)\right )}{128 d^3}-\frac{b x^3 \left (c+d x^2\right )^{3/2} (5 b c-16 a d)}{48 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d} \]

[Out]

(c*(16*a^2*d^2 + b*c*(5*b*c - 16*a*d))*x*Sqrt[c + d*x^2])/(128*d^3) + ((16*a^2*d
^2 + b*c*(5*b*c - 16*a*d))*x^3*Sqrt[c + d*x^2])/(64*d^2) - (b*(5*b*c - 16*a*d)*x
^3*(c + d*x^2)^(3/2))/(48*d^2) + (b^2*x^5*(c + d*x^2)^(3/2))/(8*d) - (c^2*(16*a^
2*d^2 + b*c*(5*b*c - 16*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(7/2)
)

_______________________________________________________________________________________

Rubi [A]  time = 0.481757, antiderivative size = 188, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{c^2 \left (16 a^2 d^2+b c (5 b c-16 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{7/2}}+\frac{1}{64} x^3 \sqrt{c+d x^2} \left (16 a^2+\frac{b c (5 b c-16 a d)}{d^2}\right )+\frac{c x \sqrt{c+d x^2} \left (16 a^2 d^2+b c (5 b c-16 a d)\right )}{128 d^3}-\frac{b x^3 \left (c+d x^2\right )^{3/2} (5 b c-16 a d)}{48 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d} \]

Antiderivative was successfully verified.

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

[Out]

(c*(16*a^2*d^2 + b*c*(5*b*c - 16*a*d))*x*Sqrt[c + d*x^2])/(128*d^3) + ((16*a^2 +
(b*c*(5*b*c - 16*a*d))/d^2)*x^3*Sqrt[c + d*x^2])/64 - (b*(5*b*c - 16*a*d)*x^3*(
c + d*x^2)^(3/2))/(48*d^2) + (b^2*x^5*(c + d*x^2)^(3/2))/(8*d) - (c^2*(16*a^2*d^
2 + b*c*(5*b*c - 16*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(128*d^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.2692, size = 182, normalized size = 0.95 \[ \frac{b^{2} x^{5} \left (c + d x^{2}\right )^{\frac{3}{2}}}{8 d} + \frac{b x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (16 a d - 5 b c\right )}{48 d^{2}} - \frac{c^{2} \left (16 a^{2} d^{2} - b c \left (16 a d - 5 b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{128 d^{\frac{7}{2}}} + \frac{c x \sqrt{c + d x^{2}} \left (16 a^{2} d^{2} - b c \left (16 a d - 5 b c\right )\right )}{128 d^{3}} + \frac{x^{3} \sqrt{c + d x^{2}} \left (16 a^{2} d^{2} - b c \left (16 a d - 5 b c\right )\right )}{64 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.160128, size = 157, normalized size = 0.82 \[ \frac{\sqrt{d} x \sqrt{c+d x^2} \left (48 a^2 d^2 \left (c+2 d x^2\right )+16 a b d \left (-3 c^2+2 c d x^2+8 d^2 x^4\right )+b^2 \left (15 c^3-10 c^2 d x^2+8 c d^2 x^4+48 d^3 x^6\right )\right )-3 c^2 \left (16 a^2 d^2-16 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{384 d^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(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*(c + 2*d*x^2) + 16*a*b*d*(-3*c^2 + 2*c*d*
x^2 + 8*d^2*x^4) + b^2*(15*c^3 - 10*c^2*d*x^2 + 8*c*d^2*x^4 + 48*d^3*x^6)) - 3*c
^2*(5*b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(38
4*d^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 259, normalized size = 1.4 \[{\frac{{a}^{2}x}{4\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}cx}{8\,d}\sqrt{d{x}^{2}+c}}-{\frac{{a}^{2}{c}^{2}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{5}}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}c{x}^{3}}{48\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}{c}^{2}x}{64\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x{b}^{2}{c}^{3}}{128\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{5\,{b}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}+{\frac{ab{x}^{3}}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{abcx}{4\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{ab{c}^{2}x}{8\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{ab{c}^{3}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^2*x*(d*x^2+c)^(3/2)/d-1/8*a^2*c/d*x*(d*x^2+c)^(1/2)-1/8*a^2*c^2/d^(3/2)*ln
(x*d^(1/2)+(d*x^2+c)^(1/2))+1/8*b^2*x^5*(d*x^2+c)^(3/2)/d-5/48*b^2*c/d^2*x^3*(d*
x^2+c)^(3/2)+5/64*b^2*c^2/d^3*x*(d*x^2+c)^(3/2)-5/128*b^2*c^3/d^3*x*(d*x^2+c)^(1
/2)-5/128*b^2*c^4/d^(7/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/3*a*b*x^3*(d*x^2+c)^(3
/2)/d-1/4*a*b*c/d^2*x*(d*x^2+c)^(3/2)+1/8*a*b*c^2/d^2*x*(d*x^2+c)^(1/2)+1/8*a*b*
c^3/d^(5/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))

_______________________________________________________________________________________

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*sqrt(d*x^2 + c)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.32494, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, b^{2} d^{3} x^{7} + 8 \,{\left (b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{5} - 2 \,{\left (5 \, b^{2} c^{2} d - 16 \, a b c d^{2} - 48 \, a^{2} d^{3}\right )} x^{3} + 3 \,{\left (5 \, b^{2} c^{3} - 16 \, a b c^{2} d + 16 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (5 \, b^{2} c^{4} - 16 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{768 \, d^{\frac{7}{2}}}, \frac{{\left (48 \, b^{2} d^{3} x^{7} + 8 \,{\left (b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{5} - 2 \,{\left (5 \, b^{2} c^{2} d - 16 \, a b c d^{2} - 48 \, a^{2} d^{3}\right )} x^{3} + 3 \,{\left (5 \, b^{2} c^{3} - 16 \, a b c^{2} d + 16 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} - 3 \,{\left (5 \, b^{2} c^{4} - 16 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{384 \, \sqrt{-d} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 57.1382, size = 411, normalized size = 2.15 \[ \frac{a^{2} c^{\frac{3}{2}} x}{8 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} \sqrt{c} x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{3}{2}}} + \frac{a^{2} d x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{\frac{5}{2}} x}{8 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{\frac{3}{2}} x^{3}}{24 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b \sqrt{c} x^{5}}{12 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{a b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 d^{\frac{5}{2}}} + \frac{a b d x^{7}}{3 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{7}{2}} x}{128 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{5}{2}} x^{3}}{384 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{3}{2}} x^{5}}{192 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{7 b^{2} \sqrt{c} x^{7}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 b^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{7}{2}}} + \frac{b^{2} d x^{9}}{8 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.247241, size = 235, normalized size = 1.23 \[ \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, b^{2} x^{2} + \frac{b^{2} c d^{5} + 16 \, a b d^{6}}{d^{6}}\right )} x^{2} - \frac{5 \, b^{2} c^{2} d^{4} - 16 \, a b c d^{5} - 48 \, a^{2} d^{6}}{d^{6}}\right )} x^{2} + \frac{3 \,{\left (5 \, b^{2} c^{3} d^{3} - 16 \, a b c^{2} d^{4} + 16 \, a^{2} c d^{5}\right )}}{d^{6}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (5 \, b^{2} c^{4} - 16 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{128 \, d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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