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

Optimal. Leaf size=281 \[ \frac{c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{9/2}}-\frac{c^3 x \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac{c^2 x^3 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}+\frac{x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{192 d^2}+\frac{c x^5 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac{b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac{b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \]

[Out]

-(c^3*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/(1024*d^4) + (c^2*(
24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x^3*Sqrt[c + d*x^2])/(1536*d^3) + (c*(24*a^2*
d^2 + b*c*(7*b*c - 24*a*d))*x^5*Sqrt[c + d*x^2])/(384*d^2) + ((24*a^2*d^2 + b*c*
(7*b*c - 24*a*d))*x^5*(c + d*x^2)^(3/2))/(192*d^2) - (b*(7*b*c - 24*a*d)*x^5*(c
+ d*x^2)^(5/2))/(120*d^2) + (b^2*x^7*(c + d*x^2)^(5/2))/(12*d) + (c^4*(24*a^2*d^
2 + b*c*(7*b*c - 24*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(1024*d^(9/2))

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Rubi [A]  time = 0.669416, 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 \[ \frac{c^4 \left (24 a^2 d^2+b c (7 b c-24 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{1024 d^{9/2}}-\frac{c^3 x \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1024 d^4}+\frac{c^2 x^3 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{1536 d^3}+\frac{1}{192} x^5 \left (c+d x^2\right )^{3/2} \left (24 a^2+\frac{b c (7 b c-24 a d)}{d^2}\right )+\frac{c x^5 \sqrt{c+d x^2} \left (24 a^2 d^2+b c (7 b c-24 a d)\right )}{384 d^2}-\frac{b x^5 \left (c+d x^2\right )^{5/2} (7 b c-24 a d)}{120 d^2}+\frac{b^2 x^7 \left (c+d x^2\right )^{5/2}}{12 d} \]

Antiderivative was successfully verified.

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

[Out]

-(c^3*(24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x*Sqrt[c + d*x^2])/(1024*d^4) + (c^2*(
24*a^2*d^2 + b*c*(7*b*c - 24*a*d))*x^3*Sqrt[c + d*x^2])/(1536*d^3) + (c*(24*a^2*
d^2 + b*c*(7*b*c - 24*a*d))*x^5*Sqrt[c + d*x^2])/(384*d^2) + ((24*a^2 + (b*c*(7*
b*c - 24*a*d))/d^2)*x^5*(c + d*x^2)^(3/2))/192 - (b*(7*b*c - 24*a*d)*x^5*(c + d*
x^2)^(5/2))/(120*d^2) + (b^2*x^7*(c + d*x^2)^(5/2))/(12*d) + (c^4*(24*a^2*d^2 +
b*c*(7*b*c - 24*a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(1024*d^(9/2))

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Rubi in Sympy [A]  time = 49.2156, size = 270, normalized size = 0.96 \[ \frac{b^{2} x^{7} \left (c + d x^{2}\right )^{\frac{5}{2}}}{12 d} + \frac{b x^{5} \left (c + d x^{2}\right )^{\frac{5}{2}} \left (24 a d - 7 b c\right )}{120 d^{2}} + \frac{c^{4} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{1024 d^{\frac{9}{2}}} - \frac{c^{3} x \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{1024 d^{4}} + \frac{c^{2} x^{3} \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{1536 d^{3}} + \frac{c x^{5} \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{384 d^{2}} + \frac{x^{5} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (24 a^{2} d^{2} - b c \left (24 a d - 7 b c\right )\right )}{192 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**7*(c + d*x**2)**(5/2)/(12*d) + b*x**5*(c + d*x**2)**(5/2)*(24*a*d - 7*b*
c)/(120*d**2) + c**4*(24*a**2*d**2 - b*c*(24*a*d - 7*b*c))*atanh(sqrt(d)*x/sqrt(
c + d*x**2))/(1024*d**(9/2)) - c**3*x*sqrt(c + d*x**2)*(24*a**2*d**2 - b*c*(24*a
*d - 7*b*c))/(1024*d**4) + c**2*x**3*sqrt(c + d*x**2)*(24*a**2*d**2 - b*c*(24*a*
d - 7*b*c))/(1536*d**3) + c*x**5*sqrt(c + d*x**2)*(24*a**2*d**2 - b*c*(24*a*d -
7*b*c))/(384*d**2) + x**5*(c + d*x**2)**(3/2)*(24*a**2*d**2 - b*c*(24*a*d - 7*b*
c))/(192*d**2)

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Mathematica [A]  time = 0.244492, size = 225, normalized size = 0.8 \[ \frac{15 c^4 \left (24 a^2 d^2-24 a b c d+7 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{d} x \sqrt{c+d x^2} \left (120 a^2 d^2 \left (-3 c^3+2 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right )+24 a b d \left (15 c^4-10 c^3 d x^2+8 c^2 d^2 x^4+176 c d^3 x^6+128 d^4 x^8\right )+b^2 \left (-105 c^5+70 c^4 d x^2-56 c^3 d^2 x^4+48 c^2 d^3 x^6+1664 c d^4 x^8+1280 d^5 x^{10}\right )\right )}{15360 d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d]*x*Sqrt[c + d*x^2]*(120*a^2*d^2*(-3*c^3 + 2*c^2*d*x^2 + 24*c*d^2*x^4 + 1
6*d^3*x^6) + 24*a*b*d*(15*c^4 - 10*c^3*d*x^2 + 8*c^2*d^2*x^4 + 176*c*d^3*x^6 + 1
28*d^4*x^8) + b^2*(-105*c^5 + 70*c^4*d*x^2 - 56*c^3*d^2*x^4 + 48*c^2*d^3*x^6 + 1
664*c*d^4*x^8 + 1280*d^5*x^10)) + 15*c^4*(7*b^2*c^2 - 24*a*b*c*d + 24*a^2*d^2)*L
og[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(15360*d^(9/2))

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Maple [A]  time = 0.03, size = 389, normalized size = 1.4 \[{\frac{{a}^{2}{x}^{3}}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}cx}{16\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}{c}^{2}x}{64\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}{c}^{3}x}{128\,{d}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}{x}^{7}}{12\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{b}^{2}c{x}^{5}}{120\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{c}^{2}{x}^{3}}{192\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{7\,x{b}^{2}{c}^{3}}{384\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{c}^{4}x}{1536\,{d}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}{c}^{5}x}{1024\,{d}^{4}}\sqrt{d{x}^{2}+c}}+{\frac{7\,{b}^{2}{c}^{6}}{1024}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{9}{2}}}}+{\frac{ab{x}^{5}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abc{x}^{3}}{8\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{c}^{2}x}{16\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{ab{c}^{3}x}{64\,{d}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ab{c}^{4}x}{128\,{d}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{3\,ab{c}^{5}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*a^2*x^3*(d*x^2+c)^(5/2)/d-1/16*a^2*c/d^2*x*(d*x^2+c)^(5/2)+1/64*a^2*c^2/d^2*
x*(d*x^2+c)^(3/2)+3/128*a^2*c^3/d^2*x*(d*x^2+c)^(1/2)+3/128*a^2*c^4/d^(5/2)*ln(x
*d^(1/2)+(d*x^2+c)^(1/2))+1/12*b^2*x^7*(d*x^2+c)^(5/2)/d-7/120*b^2*c/d^2*x^5*(d*
x^2+c)^(5/2)+7/192*b^2*c^2/d^3*x^3*(d*x^2+c)^(5/2)-7/384*b^2*c^3/d^4*x*(d*x^2+c)
^(5/2)+7/1536*b^2*c^4/d^4*x*(d*x^2+c)^(3/2)+7/1024*b^2*c^5/d^4*x*(d*x^2+c)^(1/2)
+7/1024*b^2*c^6/d^(9/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/5*a*b*x^5*(d*x^2+c)^(5/2
)/d-1/8*a*b*c/d^2*x^3*(d*x^2+c)^(5/2)+1/16*a*b*c^2/d^3*x*(d*x^2+c)^(5/2)-1/64*a*
b*c^3/d^3*x*(d*x^2+c)^(3/2)-3/128*a*b*c^4/d^3*x*(d*x^2+c)^(1/2)-3/128*a*b*c^5/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*(d*x^2 + c)^(3/2)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.83707, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, b^{2} d^{5} x^{11} + 128 \,{\left (13 \, b^{2} c d^{4} + 24 \, a b d^{5}\right )} x^{9} + 48 \,{\left (b^{2} c^{2} d^{3} + 88 \, a b c d^{4} + 40 \, a^{2} d^{5}\right )} x^{7} - 8 \,{\left (7 \, b^{2} c^{3} d^{2} - 24 \, a b c^{2} d^{3} - 360 \, a^{2} c d^{4}\right )} x^{5} + 10 \,{\left (7 \, b^{2} c^{4} d - 24 \, a b c^{3} d^{2} + 24 \, a^{2} c^{2} d^{3}\right )} x^{3} - 15 \,{\left (7 \, b^{2} c^{5} - 24 \, a b c^{4} d + 24 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 15 \,{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{30720 \, d^{\frac{9}{2}}}, \frac{{\left (1280 \, b^{2} d^{5} x^{11} + 128 \,{\left (13 \, b^{2} c d^{4} + 24 \, a b d^{5}\right )} x^{9} + 48 \,{\left (b^{2} c^{2} d^{3} + 88 \, a b c d^{4} + 40 \, a^{2} d^{5}\right )} x^{7} - 8 \,{\left (7 \, b^{2} c^{3} d^{2} - 24 \, a b c^{2} d^{3} - 360 \, a^{2} c d^{4}\right )} x^{5} + 10 \,{\left (7 \, b^{2} c^{4} d - 24 \, a b c^{3} d^{2} + 24 \, a^{2} c^{2} d^{3}\right )} x^{3} - 15 \,{\left (7 \, b^{2} c^{5} - 24 \, a b c^{4} d + 24 \, a^{2} c^{3} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} + 15 \,{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{15360 \, \sqrt{-d} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(2*(1280*b^2*d^5*x^11 + 128*(13*b^2*c*d^4 + 24*a*b*d^5)*x^9 + 48*(b^2*c
^2*d^3 + 88*a*b*c*d^4 + 40*a^2*d^5)*x^7 - 8*(7*b^2*c^3*d^2 - 24*a*b*c^2*d^3 - 36
0*a^2*c*d^4)*x^5 + 10*(7*b^2*c^4*d - 24*a*b*c^3*d^2 + 24*a^2*c^2*d^3)*x^3 - 15*(
7*b^2*c^5 - 24*a*b*c^4*d + 24*a^2*c^3*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) + 15*(7*b^
2*c^6 - 24*a*b*c^5*d + 24*a^2*c^4*d^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c
)*sqrt(d)))/d^(9/2), 1/15360*((1280*b^2*d^5*x^11 + 128*(13*b^2*c*d^4 + 24*a*b*d^
5)*x^9 + 48*(b^2*c^2*d^3 + 88*a*b*c*d^4 + 40*a^2*d^5)*x^7 - 8*(7*b^2*c^3*d^2 - 2
4*a*b*c^2*d^3 - 360*a^2*c*d^4)*x^5 + 10*(7*b^2*c^4*d - 24*a*b*c^3*d^2 + 24*a^2*c
^2*d^3)*x^3 - 15*(7*b^2*c^5 - 24*a*b*c^4*d + 24*a^2*c^3*d^2)*x)*sqrt(d*x^2 + c)*
sqrt(-d) + 15*(7*b^2*c^6 - 24*a*b*c^5*d + 24*a^2*c^4*d^2)*arctan(sqrt(-d)*x/sqrt
(d*x^2 + c)))/(sqrt(-d)*d^4)]

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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**4*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246887, size = 355, normalized size = 1.26 \[ \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, b^{2} d x^{2} + \frac{13 \, b^{2} c d^{10} + 24 \, a b d^{11}}{d^{10}}\right )} x^{2} + \frac{3 \,{\left (b^{2} c^{2} d^{9} + 88 \, a b c d^{10} + 40 \, a^{2} d^{11}\right )}}{d^{10}}\right )} x^{2} - \frac{7 \, b^{2} c^{3} d^{8} - 24 \, a b c^{2} d^{9} - 360 \, a^{2} c d^{10}}{d^{10}}\right )} x^{2} + \frac{5 \,{\left (7 \, b^{2} c^{4} d^{7} - 24 \, a b c^{3} d^{8} + 24 \, a^{2} c^{2} d^{9}\right )}}{d^{10}}\right )} x^{2} - \frac{15 \,{\left (7 \, b^{2} c^{5} d^{6} - 24 \, a b c^{4} d^{7} + 24 \, a^{2} c^{3} d^{8}\right )}}{d^{10}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (7 \, b^{2} c^{6} - 24 \, a b c^{5} d + 24 \, a^{2} c^{4} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{1024 \, d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15360*(2*(4*(2*(8*(10*b^2*d*x^2 + (13*b^2*c*d^10 + 24*a*b*d^11)/d^10)*x^2 + 3*
(b^2*c^2*d^9 + 88*a*b*c*d^10 + 40*a^2*d^11)/d^10)*x^2 - (7*b^2*c^3*d^8 - 24*a*b*
c^2*d^9 - 360*a^2*c*d^10)/d^10)*x^2 + 5*(7*b^2*c^4*d^7 - 24*a*b*c^3*d^8 + 24*a^2
*c^2*d^9)/d^10)*x^2 - 15*(7*b^2*c^5*d^6 - 24*a*b*c^4*d^7 + 24*a^2*c^3*d^8)/d^10)
*sqrt(d*x^2 + c)*x - 1/1024*(7*b^2*c^6 - 24*a*b*c^5*d + 24*a^2*c^4*d^2)*ln(abs(-
sqrt(d)*x + sqrt(d*x^2 + c)))/d^(9/2)