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

Optimal. Leaf size=197 \[ -\frac{c \left (24 a^2 d^2-60 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{9/2}}+\frac{x \sqrt{c+d x^2} \left (24 a^2 d^2-60 a b c d+35 b^2 c^2\right )}{16 d^4}-\frac{x^3 \sqrt{c+d x^2} \left (24 a^2 d^2-60 a b c d+35 b^2 c^2\right )}{24 c d^3}+\frac{x^5 (b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 x^5 \sqrt{c+d x^2}}{6 d^2} \]

[Out]

((b*c - a*d)^2*x^5)/(c*d^2*Sqrt[c + d*x^2]) + ((35*b^2*c^2 - 60*a*b*c*d + 24*a^2
*d^2)*x*Sqrt[c + d*x^2])/(16*d^4) - ((35*b^2*c^2 - 60*a*b*c*d + 24*a^2*d^2)*x^3*
Sqrt[c + d*x^2])/(24*c*d^3) + (b^2*x^5*Sqrt[c + d*x^2])/(6*d^2) - (c*(35*b^2*c^2
- 60*a*b*c*d + 24*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(9/2))

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Rubi [A]  time = 0.416537, antiderivative size = 197, 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 \[ -\frac{c \left (24 a^2 d^2-60 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 d^{9/2}}+\frac{x \sqrt{c+d x^2} \left (24 a^2 d^2-60 a b c d+35 b^2 c^2\right )}{16 d^4}-\frac{x^3 \sqrt{c+d x^2} \left (24 a^2 d^2-60 a b c d+35 b^2 c^2\right )}{24 c d^3}+\frac{x^5 (b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 x^5 \sqrt{c+d x^2}}{6 d^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*x^5)/(c*d^2*Sqrt[c + d*x^2]) + ((35*b^2*c^2 - 60*a*b*c*d + 24*a^2
*d^2)*x*Sqrt[c + d*x^2])/(16*d^4) - ((35*b^2*c^2 - 60*a*b*c*d + 24*a^2*d^2)*x^3*
Sqrt[c + d*x^2])/(24*c*d^3) + (b^2*x^5*Sqrt[c + d*x^2])/(6*d^2) - (c*(35*b^2*c^2
- 60*a*b*c*d + 24*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(16*d^(9/2))

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Rubi in Sympy [A]  time = 45.9704, size = 189, normalized size = 0.96 \[ \frac{b^{2} x^{5} \sqrt{c + d x^{2}}}{6 d^{2}} - \frac{c \left (24 a^{2} d^{2} - 60 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 d^{\frac{9}{2}}} + \frac{x \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - 60 a b c d + 35 b^{2} c^{2}\right )}{16 d^{4}} + \frac{x^{5} \left (a d - b c\right )^{2}}{c d^{2} \sqrt{c + d x^{2}}} - \frac{x^{3} \sqrt{c + d x^{2}} \left (24 a^{2} d^{2} - 60 a b c d + 35 b^{2} c^{2}\right )}{24 c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**5*sqrt(c + d*x**2)/(6*d**2) - c*(24*a**2*d**2 - 60*a*b*c*d + 35*b**2*c**
2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(16*d**(9/2)) + x*sqrt(c + d*x**2)*(24*a**2
*d**2 - 60*a*b*c*d + 35*b**2*c**2)/(16*d**4) + x**5*(a*d - b*c)**2/(c*d**2*sqrt(
c + d*x**2)) - x**3*sqrt(c + d*x**2)*(24*a**2*d**2 - 60*a*b*c*d + 35*b**2*c**2)/
(24*c*d**3)

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Mathematica [A]  time = 0.256648, size = 158, normalized size = 0.8 \[ \sqrt{c+d x^2} \left (\frac{x \left (8 a^2 d^2-28 a b c d+19 b^2 c^2\right )}{16 d^4}+\frac{c x (b c-a d)^2}{d^4 \left (c+d x^2\right )}-\frac{b x^3 (11 b c-12 a d)}{24 d^3}+\frac{b^2 x^5}{6 d^2}\right )-\frac{c \left (24 a^2 d^2-60 a b c d+35 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{16 d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x^2]*(((19*b^2*c^2 - 28*a*b*c*d + 8*a^2*d^2)*x)/(16*d^4) - (b*(11*b*c
- 12*a*d)*x^3)/(24*d^3) + (b^2*x^5)/(6*d^2) + (c*(b*c - a*d)^2*x)/(d^4*(c + d*x
^2))) - (c*(35*b^2*c^2 - 60*a*b*c*d + 24*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x
^2]])/(16*d^(9/2))

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Maple [A]  time = 0.025, size = 263, normalized size = 1.3 \[{\frac{{a}^{2}{x}^{3}}{2\,d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{3\,{a}^{2}cx}{2\,{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{3\,{a}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}{x}^{7}}{6\,d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{7\,{b}^{2}c{x}^{5}}{24\,{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{35\,{b}^{2}{c}^{2}{x}^{3}}{48\,{d}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{35\,x{b}^{2}{c}^{3}}{16\,{d}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{35\,{b}^{2}{c}^{3}}{16}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{9}{2}}}}+{\frac{ab{x}^{5}}{2\,d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{5\,abc{x}^{3}}{4\,{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{15\,ab{c}^{2}x}{4\,{d}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{15\,ab{c}^{2}}{4}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{7}{2}}}} \]

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]

1/2*a^2*x^3/d/(d*x^2+c)^(1/2)+3/2*a^2*c/d^2*x/(d*x^2+c)^(1/2)-3/2*a^2*c/d^(5/2)*
ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/6*b^2*x^7/d/(d*x^2+c)^(1/2)-7/24*b^2*c/d^2*x^5/(
d*x^2+c)^(1/2)+35/48*b^2*c^2/d^3*x^3/(d*x^2+c)^(1/2)+35/16*b^2*c^3/d^4*x/(d*x^2+
c)^(1/2)-35/16*b^2*c^3/d^(9/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/2*a*b*x^5/d/(d*x^
2+c)^(1/2)-5/4*a*b*c/d^2*x^3/(d*x^2+c)^(1/2)-15/4*a*b*c^2/d^3*x/(d*x^2+c)^(1/2)+
15/4*a*b*c^2/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/(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31597, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, b^{2} d^{3} x^{7} - 2 \,{\left (7 \, b^{2} c d^{2} - 12 \, a b d^{3}\right )} x^{5} +{\left (35 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 24 \, a^{2} d^{3}\right )} x^{3} + 3 \,{\left (35 \, b^{2} c^{3} - 60 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} + 3 \,{\left (35 \, b^{2} c^{4} - 60 \, a b c^{3} d + 24 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 24 \, a^{2} c d^{3}\right )} x^{2}\right )} \log \left (2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{96 \,{\left (d^{5} x^{2} + c d^{4}\right )} \sqrt{d}}, \frac{{\left (8 \, b^{2} d^{3} x^{7} - 2 \,{\left (7 \, b^{2} c d^{2} - 12 \, a b d^{3}\right )} x^{5} +{\left (35 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 24 \, a^{2} d^{3}\right )} x^{3} + 3 \,{\left (35 \, b^{2} c^{3} - 60 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} - 3 \,{\left (35 \, b^{2} c^{4} - 60 \, a b c^{3} d + 24 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 24 \, a^{2} c d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{48 \,{\left (d^{5} x^{2} + c d^{4}\right )} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*b^2*d^3*x^7 - 2*(7*b^2*c*d^2 - 12*a*b*d^3)*x^5 + (35*b^2*c^2*d - 60*
a*b*c*d^2 + 24*a^2*d^3)*x^3 + 3*(35*b^2*c^3 - 60*a*b*c^2*d + 24*a^2*c*d^2)*x)*sq
rt(d*x^2 + c)*sqrt(d) + 3*(35*b^2*c^4 - 60*a*b*c^3*d + 24*a^2*c^2*d^2 + (35*b^2*
c^3*d - 60*a*b*c^2*d^2 + 24*a^2*c*d^3)*x^2)*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^2
+ c)*sqrt(d)))/((d^5*x^2 + c*d^4)*sqrt(d)), 1/48*((8*b^2*d^3*x^7 - 2*(7*b^2*c*d
^2 - 12*a*b*d^3)*x^5 + (35*b^2*c^2*d - 60*a*b*c*d^2 + 24*a^2*d^3)*x^3 + 3*(35*b^
2*c^3 - 60*a*b*c^2*d + 24*a^2*c*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d) - 3*(35*b^2*c^4
- 60*a*b*c^3*d + 24*a^2*c^2*d^2 + (35*b^2*c^3*d - 60*a*b*c^2*d^2 + 24*a^2*c*d^3
)*x^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/((d^5*x^2 + c*d^4)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

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]

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

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GIAC/XCAS [A]  time = 0.241186, size = 236, normalized size = 1.2 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \, b^{2} x^{2}}{d} - \frac{7 \, b^{2} c d^{5} - 12 \, a b d^{6}}{d^{7}}\right )} x^{2} + \frac{35 \, b^{2} c^{2} d^{4} - 60 \, a b c d^{5} + 24 \, a^{2} d^{6}}{d^{7}}\right )} x^{2} + \frac{3 \,{\left (35 \, b^{2} c^{3} d^{3} - 60 \, a b c^{2} d^{4} + 24 \, a^{2} c d^{5}\right )}}{d^{7}}\right )} x}{48 \, \sqrt{d x^{2} + c}} + \frac{{\left (35 \, b^{2} c^{3} - 60 \, a b c^{2} d + 24 \, a^{2} c d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{16 \, d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*((2*(4*b^2*x^2/d - (7*b^2*c*d^5 - 12*a*b*d^6)/d^7)*x^2 + (35*b^2*c^2*d^4 -
60*a*b*c*d^5 + 24*a^2*d^6)/d^7)*x^2 + 3*(35*b^2*c^3*d^3 - 60*a*b*c^2*d^4 + 24*a^
2*c*d^5)/d^7)*x/sqrt(d*x^2 + c) + 1/16*(35*b^2*c^3 - 60*a*b*c^2*d + 24*a^2*c*d^2
)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(9/2)