3.626 \(\int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=281 \[ -\frac{c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{7/2}}+\frac{c^3 x \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{1024 d^3}+\frac{c^2 x^3 \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{512 d^2}+\frac{x^3 \left (c+d x^2\right )^{5/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{320 d^2}+\frac{c x^3 \left (c+d x^2\right )^{3/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{384 d^2}-\frac{b x^3 \left (c+d x^2\right )^{7/2} (5 b c-24 a d)}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d} \]

[Out]

(c^3*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/(1024*d^3) + (c^2*(4
0*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x^3*Sqrt[c + d*x^2])/(512*d^2) + (c*(40*a^2*d^
2 + b*c*(5*b*c - 24*a*d))*x^3*(c + d*x^2)^(3/2))/(384*d^2) + ((40*a^2*d^2 + b*c*
(5*b*c - 24*a*d))*x^3*(c + d*x^2)^(5/2))/(320*d^2) - (b*(5*b*c - 24*a*d)*x^3*(c
+ d*x^2)^(7/2))/(120*d^2) + (b^2*x^5*(c + d*x^2)^(7/2))/(12*d) - (c^4*(40*a^2*d^
2 + b*c*(5*b*c - 24*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(1024*d^(7/2))

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Rubi [A]  time = 0.652653, antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{c^4 \left (40 a^2 d^2+b c (5 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{7/2}}+\frac{c^3 x \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{1024 d^3}+\frac{c^2 x^3 \sqrt{c+d x^2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{512 d^2}+\frac{1}{320} x^3 \left (c+d x^2\right )^{5/2} \left (40 a^2+\frac{b c (5 b c-24 a d)}{d^2}\right )+\frac{c x^3 \left (c+d x^2\right )^{3/2} \left (40 a^2 d^2+b c (5 b c-24 a d)\right )}{384 d^2}-\frac{b x^3 \left (c+d x^2\right )^{7/2} (5 b c-24 a d)}{120 d^2}+\frac{b^2 x^5 \left (c+d x^2\right )^{7/2}}{12 d} \]

Antiderivative was successfully verified.

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

[Out]

(c^3*(40*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/(1024*d^3) + (c^2*(4
0*a^2*d^2 + b*c*(5*b*c - 24*a*d))*x^3*Sqrt[c + d*x^2])/(512*d^2) + (c*(40*a^2*d^
2 + b*c*(5*b*c - 24*a*d))*x^3*(c + d*x^2)^(3/2))/(384*d^2) + ((40*a^2 + (b*c*(5*
b*c - 24*a*d))/d^2)*x^3*(c + d*x^2)^(5/2))/320 - (b*(5*b*c - 24*a*d)*x^3*(c + d*
x^2)^(7/2))/(120*d^2) + (b^2*x^5*(c + d*x^2)^(7/2))/(12*d) - (c^4*(40*a^2*d^2 +
b*c*(5*b*c - 24*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(1024*d^(7/2))

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Rubi in Sympy [A]  time = 53.6021, size = 270, normalized size = 0.96 \[ \frac{b^{2} x^{5} \left (c + d x^{2}\right )^{\frac{7}{2}}}{12 d} + \frac{b x^{3} \left (c + d x^{2}\right )^{\frac{7}{2}} \left (24 a d - 5 b c\right )}{120 d^{2}} - \frac{c^{4} \left (40 a^{2} d^{2} - b c \left (24 a d - 5 b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{1024 d^{\frac{7}{2}}} + \frac{c^{3} x \sqrt{c + d x^{2}} \left (40 a^{2} d^{2} - b c \left (24 a d - 5 b c\right )\right )}{1024 d^{3}} + \frac{c^{2} x^{3} \sqrt{c + d x^{2}} \left (40 a^{2} d^{2} - b c \left (24 a d - 5 b c\right )\right )}{512 d^{2}} + \frac{c x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (40 a^{2} d^{2} - b c \left (24 a d - 5 b c\right )\right )}{384 d^{2}} + \frac{x^{3} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (40 a^{2} d^{2} - b c \left (24 a d - 5 b c\right )\right )}{320 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**5*(c + d*x**2)**(7/2)/(12*d) + b*x**3*(c + d*x**2)**(7/2)*(24*a*d - 5*b*
c)/(120*d**2) - c**4*(40*a**2*d**2 - b*c*(24*a*d - 5*b*c))*atanh(sqrt(d)*x/sqrt(
c + d*x**2))/(1024*d**(7/2)) + c**3*x*sqrt(c + d*x**2)*(40*a**2*d**2 - b*c*(24*a
*d - 5*b*c))/(1024*d**3) + c**2*x**3*sqrt(c + d*x**2)*(40*a**2*d**2 - b*c*(24*a*
d - 5*b*c))/(512*d**2) + c*x**3*(c + d*x**2)**(3/2)*(40*a**2*d**2 - b*c*(24*a*d
- 5*b*c))/(384*d**2) + x**3*(c + d*x**2)**(5/2)*(40*a**2*d**2 - b*c*(24*a*d - 5*
b*c))/(320*d**2)

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Mathematica [A]  time = 0.24853, size = 226, normalized size = 0.8 \[ \frac{\sqrt{d} x \sqrt{c+d x^2} \left (40 a^2 d^2 \left (15 c^3+118 c^2 d x^2+136 c d^2 x^4+48 d^3 x^6\right )+24 a b d \left (-15 c^4+10 c^3 d x^2+248 c^2 d^2 x^4+336 c d^3 x^6+128 d^4 x^8\right )+5 b^2 \left (15 c^5-10 c^4 d x^2+8 c^3 d^2 x^4+432 c^2 d^3 x^6+640 c d^4 x^8+256 d^5 x^{10}\right )\right )-15 c^4 \left (40 a^2 d^2-24 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{15360 d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(40*a^2*d^2*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 +
48*d^3*x^6) + 24*a*b*d*(-15*c^4 + 10*c^3*d*x^2 + 248*c^2*d^2*x^4 + 336*c*d^3*x^
6 + 128*d^4*x^8) + 5*b^2*(15*c^5 - 10*c^4*d*x^2 + 8*c^3*d^2*x^4 + 432*c^2*d^3*x^
6 + 640*c*d^4*x^8 + 256*d^5*x^10)) - 15*c^4*(5*b^2*c^2 - 24*a*b*c*d + 40*a^2*d^2
)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(15360*d^(7/2))

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Maple [A]  time = 0.023, size = 383, normalized size = 1.4 \[{\frac{{a}^{2}x}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}cx}{48\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}{c}^{2}x}{192\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{c}^{3}x}{128\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,{a}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{5}}{12\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}c{x}^{3}}{24\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}{c}^{2}x}{64\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{x{b}^{2}{c}^{3}}{384\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}{c}^{4}x}{1536\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}{c}^{5}x}{1024\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{5\,{b}^{2}{c}^{6}}{1024}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}}+{\frac{ab{x}^{3}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{3\,abcx}{40\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{ab{c}^{2}x}{80\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{c}^{3}x}{64\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ab{c}^{4}x}{128\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,ab{c}^{5}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*a^2*x*(d*x^2+c)^(7/2)/d-1/48*a^2*c/d*x*(d*x^2+c)^(5/2)-5/192*a^2*c^2/d*x*(d*
x^2+c)^(3/2)-5/128*a^2*c^3/d*x*(d*x^2+c)^(1/2)-5/128*a^2*c^4/d^(3/2)*ln(x*d^(1/2
)+(d*x^2+c)^(1/2))+1/12*b^2*x^5*(d*x^2+c)^(7/2)/d-1/24*b^2*c/d^2*x^3*(d*x^2+c)^(
7/2)+1/64*b^2*c^2/d^3*x*(d*x^2+c)^(7/2)-1/384*b^2*c^3/d^3*x*(d*x^2+c)^(5/2)-5/15
36*b^2*c^4/d^3*x*(d*x^2+c)^(3/2)-5/1024*b^2*c^5/d^3*x*(d*x^2+c)^(1/2)-5/1024*b^2
*c^6/d^(7/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/5*a*b*x^3*(d*x^2+c)^(7/2)/d-3/40*a*
b*c/d^2*x*(d*x^2+c)^(7/2)+1/80*a*b*c^2/d^2*x*(d*x^2+c)^(5/2)+1/64*a*b*c^3/d^2*x*
(d*x^2+c)^(3/2)+3/128*a*b*c^4/d^2*x*(d*x^2+c)^(1/2)+3/128*a*b*c^5/d^(5/2)*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*(d*x^2 + c)^(5/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.827388, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, b^{2} d^{5} x^{11} + 128 \,{\left (25 \, b^{2} c d^{4} + 24 \, a b d^{5}\right )} x^{9} + 48 \,{\left (45 \, b^{2} c^{2} d^{3} + 168 \, a b c d^{4} + 40 \, a^{2} d^{5}\right )} x^{7} + 8 \,{\left (5 \, b^{2} c^{3} d^{2} + 744 \, a b c^{2} d^{3} + 680 \, a^{2} c d^{4}\right )} x^{5} - 10 \,{\left (5 \, b^{2} c^{4} d - 24 \, a b c^{3} d^{2} - 472 \, a^{2} c^{2} d^{3}\right )} x^{3} + 15 \,{\left (5 \, b^{2} c^{5} - 24 \, a b c^{4} d + 40 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 15 \,{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{30720 \, d^{\frac{7}{2}}}, \frac{{\left (1280 \, b^{2} d^{5} x^{11} + 128 \,{\left (25 \, b^{2} c d^{4} + 24 \, a b d^{5}\right )} x^{9} + 48 \,{\left (45 \, b^{2} c^{2} d^{3} + 168 \, a b c d^{4} + 40 \, a^{2} d^{5}\right )} x^{7} + 8 \,{\left (5 \, b^{2} c^{3} d^{2} + 744 \, a b c^{2} d^{3} + 680 \, a^{2} c d^{4}\right )} x^{5} - 10 \,{\left (5 \, b^{2} c^{4} d - 24 \, a b c^{3} d^{2} - 472 \, a^{2} c^{2} d^{3}\right )} x^{3} + 15 \,{\left (5 \, b^{2} c^{5} - 24 \, a b c^{4} d + 40 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} - 15 \,{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{15360 \, \sqrt{-d} d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(2*(1280*b^2*d^5*x^11 + 128*(25*b^2*c*d^4 + 24*a*b*d^5)*x^9 + 48*(45*b^
2*c^2*d^3 + 168*a*b*c*d^4 + 40*a^2*d^5)*x^7 + 8*(5*b^2*c^3*d^2 + 744*a*b*c^2*d^3
+ 680*a^2*c*d^4)*x^5 - 10*(5*b^2*c^4*d - 24*a*b*c^3*d^2 - 472*a^2*c^2*d^3)*x^3
+ 15*(5*b^2*c^5 - 24*a*b*c^4*d + 40*a^2*c^3*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) + 15
*(5*b^2*c^6 - 24*a*b*c^5*d + 40*a^2*c^4*d^2)*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^
2 + c)*sqrt(d)))/d^(7/2), 1/15360*((1280*b^2*d^5*x^11 + 128*(25*b^2*c*d^4 + 24*a
*b*d^5)*x^9 + 48*(45*b^2*c^2*d^3 + 168*a*b*c*d^4 + 40*a^2*d^5)*x^7 + 8*(5*b^2*c^
3*d^2 + 744*a*b*c^2*d^3 + 680*a^2*c*d^4)*x^5 - 10*(5*b^2*c^4*d - 24*a*b*c^3*d^2
- 472*a^2*c^2*d^3)*x^3 + 15*(5*b^2*c^5 - 24*a*b*c^4*d + 40*a^2*c^3*d^2)*x)*sqrt(
d*x^2 + c)*sqrt(-d) - 15*(5*b^2*c^6 - 24*a*b*c^5*d + 40*a^2*c^4*d^2)*arctan(sqrt
(-d)*x/sqrt(d*x^2 + c)))/(sqrt(-d)*d^3)]

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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(x**2*(b*x**2+a)**2*(d*x**2+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.241885, size = 358, normalized size = 1.27 \[ \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, b^{2} d^{2} x^{2} + \frac{25 \, b^{2} c d^{11} + 24 \, a b d^{12}}{d^{10}}\right )} x^{2} + \frac{3 \,{\left (45 \, b^{2} c^{2} d^{10} + 168 \, a b c d^{11} + 40 \, a^{2} d^{12}\right )}}{d^{10}}\right )} x^{2} + \frac{5 \, b^{2} c^{3} d^{9} + 744 \, a b c^{2} d^{10} + 680 \, a^{2} c d^{11}}{d^{10}}\right )} x^{2} - \frac{5 \,{\left (5 \, b^{2} c^{4} d^{8} - 24 \, a b c^{3} d^{9} - 472 \, a^{2} c^{2} d^{10}\right )}}{d^{10}}\right )} x^{2} + \frac{15 \,{\left (5 \, b^{2} c^{5} d^{7} - 24 \, a b c^{4} d^{8} + 40 \, a^{2} c^{3} d^{9}\right )}}{d^{10}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (5 \, b^{2} c^{6} - 24 \, a b c^{5} d + 40 \, a^{2} c^{4} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{1024 \, d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360*(2*(4*(2*(8*(10*b^2*d^2*x^2 + (25*b^2*c*d^11 + 24*a*b*d^12)/d^10)*x^2 +
3*(45*b^2*c^2*d^10 + 168*a*b*c*d^11 + 40*a^2*d^12)/d^10)*x^2 + (5*b^2*c^3*d^9 +
744*a*b*c^2*d^10 + 680*a^2*c*d^11)/d^10)*x^2 - 5*(5*b^2*c^4*d^8 - 24*a*b*c^3*d^9
- 472*a^2*c^2*d^10)/d^10)*x^2 + 15*(5*b^2*c^5*d^7 - 24*a*b*c^4*d^8 + 40*a^2*c^3
*d^9)/d^10)*sqrt(d*x^2 + c)*x + 1/1024*(5*b^2*c^6 - 24*a*b*c^5*d + 40*a^2*c^4*d^
2)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(7/2)