3.615 \(\int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=235 \[ -\frac{c^3 \left (16 a^2 d^2+3 b c (b c-4 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{7/2}}+\frac{c^2 x \sqrt{c+d x^2} \left (16 a^2 d^2+3 b c (b c-4 a d)\right )}{256 d^3}+\frac{x^3 \left (c+d x^2\right )^{3/2} \left (16 a^2 d^2+3 b c (b c-4 a d)\right )}{96 d^2}+\frac{c x^3 \sqrt{c+d x^2} \left (16 a^2 d^2+3 b c (b c-4 a d)\right )}{128 d^2}-\frac{b x^3 \left (c+d x^2\right )^{5/2} (b c-4 a d)}{16 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{5/2}}{10 d} \]

[Out]

(c^2*(16*a^2*d^2 + 3*b*c*(b*c - 4*a*d))*x*Sqrt[c + d*x^2])/(256*d^3) + (c*(16*a^
2*d^2 + 3*b*c*(b*c - 4*a*d))*x^3*Sqrt[c + d*x^2])/(128*d^2) + ((16*a^2*d^2 + 3*b
*c*(b*c - 4*a*d))*x^3*(c + d*x^2)^(3/2))/(96*d^2) - (b*(b*c - 4*a*d)*x^3*(c + d*
x^2)^(5/2))/(16*d^2) + (b^2*x^5*(c + d*x^2)^(5/2))/(10*d) - (c^3*(16*a^2*d^2 + 3
*b*c*(b*c - 4*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(256*d^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.565347, antiderivative size = 232, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{c^3 \left (16 a^2 d^2+3 b c (b c-4 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{256 d^{7/2}}+\frac{c^2 x \sqrt{c+d x^2} \left (16 a^2 d^2+3 b c (b c-4 a d)\right )}{256 d^3}+\frac{1}{96} x^3 \left (c+d x^2\right )^{3/2} \left (16 a^2+\frac{3 b c (b c-4 a d)}{d^2}\right )+\frac{c x^3 \sqrt{c+d x^2} \left (16 a^2 d^2+3 b c (b c-4 a d)\right )}{128 d^2}-\frac{b x^3 \left (c+d x^2\right )^{5/2} (b c-4 a d)}{16 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{5/2}}{10 d} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*(16*a^2*d^2 + 3*b*c*(b*c - 4*a*d))*x*Sqrt[c + d*x^2])/(256*d^3) + (c*(16*a^
2*d^2 + 3*b*c*(b*c - 4*a*d))*x^3*Sqrt[c + d*x^2])/(128*d^2) + ((16*a^2 + (3*b*c*
(b*c - 4*a*d))/d^2)*x^3*(c + d*x^2)^(3/2))/96 - (b*(b*c - 4*a*d)*x^3*(c + d*x^2)
^(5/2))/(16*d^2) + (b^2*x^5*(c + d*x^2)^(5/2))/(10*d) - (c^3*(16*a^2*d^2 + 3*b*c
*(b*c - 4*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(256*d^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.126, size = 224, normalized size = 0.95 \[ \frac{b^{2} x^{5} \left (c + d x^{2}\right )^{\frac{5}{2}}}{10 d} + \frac{b x^{3} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (4 a d - b c\right )}{16 d^{2}} - \frac{c^{3} \left (16 a^{2} d^{2} - 3 b c \left (4 a d - b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{256 d^{\frac{7}{2}}} + \frac{c^{2} x \sqrt{c + d x^{2}} \left (16 a^{2} d^{2} - 3 b c \left (4 a d - b c\right )\right )}{256 d^{3}} + \frac{c x^{3} \sqrt{c + d x^{2}} \left (16 a^{2} d^{2} - 3 b c \left (4 a d - b c\right )\right )}{128 d^{2}} + \frac{x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (16 a^{2} d^{2} - 3 b c \left (4 a d - b c\right )\right )}{96 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**5*(c + d*x**2)**(5/2)/(10*d) + b*x**3*(c + d*x**2)**(5/2)*(4*a*d - b*c)/
(16*d**2) - c**3*(16*a**2*d**2 - 3*b*c*(4*a*d - b*c))*atanh(sqrt(d)*x/sqrt(c + d
*x**2))/(256*d**(7/2)) + c**2*x*sqrt(c + d*x**2)*(16*a**2*d**2 - 3*b*c*(4*a*d -
b*c))/(256*d**3) + c*x**3*sqrt(c + d*x**2)*(16*a**2*d**2 - 3*b*c*(4*a*d - b*c))/
(128*d**2) + x**3*(c + d*x**2)**(3/2)*(16*a**2*d**2 - 3*b*c*(4*a*d - b*c))/(96*d
**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.205001, size = 193, normalized size = 0.82 \[ \frac{\sqrt{d} x \sqrt{c+d x^2} \left (80 a^2 d^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )+60 a b d \left (-3 c^3+2 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right )+3 b^2 \left (15 c^4-10 c^3 d x^2+8 c^2 d^2 x^4+176 c d^3 x^6+128 d^4 x^8\right )\right )-15 c^3 \left (16 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{3840 d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(80*a^2*d^2*(3*c^2 + 14*c*d*x^2 + 8*d^2*x^4) + 60*a*b
*d*(-3*c^3 + 2*c^2*d*x^2 + 24*c*d^2*x^4 + 16*d^3*x^6) + 3*b^2*(15*c^4 - 10*c^3*d
*x^2 + 8*c^2*d^2*x^4 + 176*c*d^3*x^6 + 128*d^4*x^8)) - 15*c^3*(3*b^2*c^2 - 12*a*
b*c*d + 16*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(3840*d^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 321, normalized size = 1.4 \[{\frac{{a}^{2}x}{6\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}cx}{24\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{c}^{2}x}{16\,d}\sqrt{d{x}^{2}+c}}-{\frac{{a}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{5}}{10\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}c{x}^{3}}{16\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{32\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{x{b}^{2}{c}^{3}}{128\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}{c}^{4}x}{256\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{3\,{b}^{2}{c}^{5}}{256}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}+{\frac{ab{x}^{3}}{4\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abcx}{8\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{c}^{2}x}{32\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ab{c}^{3}x}{64\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,ab{c}^{4}}{64}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*a^2*x*(d*x^2+c)^(5/2)/d-1/24*a^2*c/d*x*(d*x^2+c)^(3/2)-1/16*a^2*c^2/d*x*(d*x
^2+c)^(1/2)-1/16*a^2*c^3/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/10*b^2*x^5*(d*x
^2+c)^(5/2)/d-1/16*b^2*c/d^2*x^3*(d*x^2+c)^(5/2)+1/32*b^2*c^2/d^3*x*(d*x^2+c)^(5
/2)-1/128*b^2*c^3/d^3*x*(d*x^2+c)^(3/2)-3/256*b^2*c^4/d^3*x*(d*x^2+c)^(1/2)-3/25
6*b^2*c^5/d^(7/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/4*a*b*x^3*(d*x^2+c)^(5/2)/d-1/
8*a*b*c/d^2*x*(d*x^2+c)^(5/2)+1/32*a*b*c^2/d^2*x*(d*x^2+c)^(3/2)+3/64*a*b*c^3/d^
2*x*(d*x^2+c)^(1/2)+3/64*a*b*c^4/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*(d*x^2 + c)^(3/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.497441, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (384 \, b^{2} d^{4} x^{9} + 48 \,{\left (11 \, b^{2} c d^{3} + 20 \, a b d^{4}\right )} x^{7} + 8 \,{\left (3 \, b^{2} c^{2} d^{2} + 180 \, a b c d^{3} + 80 \, a^{2} d^{4}\right )} x^{5} - 10 \,{\left (3 \, b^{2} c^{3} d - 12 \, a b c^{2} d^{2} - 112 \, a^{2} c d^{3}\right )} x^{3} + 15 \,{\left (3 \, b^{2} c^{4} - 12 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 15 \,{\left (3 \, b^{2} c^{5} - 12 \, a b c^{4} d + 16 \, a^{2} c^{3} d^{2}\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{7680 \, d^{\frac{7}{2}}}, \frac{{\left (384 \, b^{2} d^{4} x^{9} + 48 \,{\left (11 \, b^{2} c d^{3} + 20 \, a b d^{4}\right )} x^{7} + 8 \,{\left (3 \, b^{2} c^{2} d^{2} + 180 \, a b c d^{3} + 80 \, a^{2} d^{4}\right )} x^{5} - 10 \,{\left (3 \, b^{2} c^{3} d - 12 \, a b c^{2} d^{2} - 112 \, a^{2} c d^{3}\right )} x^{3} + 15 \,{\left (3 \, b^{2} c^{4} - 12 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} - 15 \,{\left (3 \, b^{2} c^{5} - 12 \, a b c^{4} d + 16 \, a^{2} c^{3} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{3840 \, \sqrt{-d} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(2*(384*b^2*d^4*x^9 + 48*(11*b^2*c*d^3 + 20*a*b*d^4)*x^7 + 8*(3*b^2*c^2*
d^2 + 180*a*b*c*d^3 + 80*a^2*d^4)*x^5 - 10*(3*b^2*c^3*d - 12*a*b*c^2*d^2 - 112*a
^2*c*d^3)*x^3 + 15*(3*b^2*c^4 - 12*a*b*c^3*d + 16*a^2*c^2*d^2)*x)*sqrt(d*x^2 + c
)*sqrt(d) + 15*(3*b^2*c^5 - 12*a*b*c^4*d + 16*a^2*c^3*d^2)*log(2*sqrt(d*x^2 + c)
*d*x - (2*d*x^2 + c)*sqrt(d)))/d^(7/2), 1/3840*((384*b^2*d^4*x^9 + 48*(11*b^2*c*
d^3 + 20*a*b*d^4)*x^7 + 8*(3*b^2*c^2*d^2 + 180*a*b*c*d^3 + 80*a^2*d^4)*x^5 - 10*
(3*b^2*c^3*d - 12*a*b*c^2*d^2 - 112*a^2*c*d^3)*x^3 + 15*(3*b^2*c^4 - 12*a*b*c^3*
d + 16*a^2*c^2*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d) - 15*(3*b^2*c^5 - 12*a*b*c^4*d +
16*a^2*c^3*d^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(sqrt(-d)*d^3)]

_______________________________________________________________________________________

Sympy [A]  time = 138.387, size = 505, normalized size = 2.15 \[ \frac{a^{2} c^{\frac{5}{2}} x}{16 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a^{2} c^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{11 a^{2} \sqrt{c} d x^{5}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{16 d^{\frac{3}{2}}} + \frac{a^{2} d^{2} x^{7}}{6 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 a b c^{\frac{7}{2}} x}{64 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a b c^{\frac{5}{2}} x^{3}}{64 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{13 a b c^{\frac{3}{2}} x^{5}}{32 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b \sqrt{c} d x^{7}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a b c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{64 d^{\frac{5}{2}}} + \frac{a b d^{2} x^{9}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} c^{\frac{9}{2}} x}{256 d^{3} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{b^{2} c^{\frac{7}{2}} x^{3}}{256 d^{2} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{b^{2} c^{\frac{5}{2}} x^{5}}{640 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 b^{2} c^{\frac{3}{2}} x^{7}}{160 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{19 b^{2} \sqrt{c} d x^{9}}{80 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{3 b^{2} c^{5} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{256 d^{\frac{7}{2}}} + \frac{b^{2} d^{2} x^{11}}{10 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**(5/2)*x/(16*d*sqrt(1 + d*x**2/c)) + 17*a**2*c**(3/2)*x**3/(48*sqrt(1 + d
*x**2/c)) + 11*a**2*sqrt(c)*d*x**5/(24*sqrt(1 + d*x**2/c)) - a**2*c**3*asinh(sqr
t(d)*x/sqrt(c))/(16*d**(3/2)) + a**2*d**2*x**7/(6*sqrt(c)*sqrt(1 + d*x**2/c)) -
3*a*b*c**(7/2)*x/(64*d**2*sqrt(1 + d*x**2/c)) - a*b*c**(5/2)*x**3/(64*d*sqrt(1 +
d*x**2/c)) + 13*a*b*c**(3/2)*x**5/(32*sqrt(1 + d*x**2/c)) + 5*a*b*sqrt(c)*d*x**
7/(8*sqrt(1 + d*x**2/c)) + 3*a*b*c**4*asinh(sqrt(d)*x/sqrt(c))/(64*d**(5/2)) + a
*b*d**2*x**9/(4*sqrt(c)*sqrt(1 + d*x**2/c)) + 3*b**2*c**(9/2)*x/(256*d**3*sqrt(1
+ d*x**2/c)) + b**2*c**(7/2)*x**3/(256*d**2*sqrt(1 + d*x**2/c)) - b**2*c**(5/2)
*x**5/(640*d*sqrt(1 + d*x**2/c)) + 23*b**2*c**(3/2)*x**7/(160*sqrt(1 + d*x**2/c)
) + 19*b**2*sqrt(c)*d*x**9/(80*sqrt(1 + d*x**2/c)) - 3*b**2*c**5*asinh(sqrt(d)*x
/sqrt(c))/(256*d**(7/2)) + b**2*d**2*x**11/(10*sqrt(c)*sqrt(1 + d*x**2/c))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229666, size = 296, normalized size = 1.26 \[ \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b^{2} d x^{2} + \frac{11 \, b^{2} c d^{8} + 20 \, a b d^{9}}{d^{8}}\right )} x^{2} + \frac{3 \, b^{2} c^{2} d^{7} + 180 \, a b c d^{8} + 80 \, a^{2} d^{9}}{d^{8}}\right )} x^{2} - \frac{5 \,{\left (3 \, b^{2} c^{3} d^{6} - 12 \, a b c^{2} d^{7} - 112 \, a^{2} c d^{8}\right )}}{d^{8}}\right )} x^{2} + \frac{15 \,{\left (3 \, b^{2} c^{4} d^{5} - 12 \, a b c^{3} d^{6} + 16 \, a^{2} c^{2} d^{7}\right )}}{d^{8}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (3 \, b^{2} c^{5} - 12 \, a b c^{4} d + 16 \, a^{2} c^{3} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{256 \, d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*(2*(4*(6*(8*b^2*d*x^2 + (11*b^2*c*d^8 + 20*a*b*d^9)/d^8)*x^2 + (3*b^2*c^2
*d^7 + 180*a*b*c*d^8 + 80*a^2*d^9)/d^8)*x^2 - 5*(3*b^2*c^3*d^6 - 12*a*b*c^2*d^7
- 112*a^2*c*d^8)/d^8)*x^2 + 15*(3*b^2*c^4*d^5 - 12*a*b*c^3*d^6 + 16*a^2*c^2*d^7)
/d^8)*sqrt(d*x^2 + c)*x + 1/256*(3*b^2*c^5 - 12*a*b*c^4*d + 16*a^2*c^3*d^2)*ln(a
bs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(7/2)