3.641 \(\int \frac{x^9}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=123 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{4 b^2 d^{3/2}}+\frac{x^2 \sqrt{c+d x^4}}{4 b d} \]

[Out]

(x^2*Sqrt[c + d*x^4])/(4*b*d) + (a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*S
qrt[c + d*x^4])])/(2*b^2*Sqrt[b*c - a*d]) - ((b*c + 2*a*d)*ArcTanh[(Sqrt[d]*x^2)
/Sqrt[c + d*x^4]])/(4*b^2*d^(3/2))

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Rubi [A]  time = 0.393122, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 b^2 \sqrt{b c-a d}}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{4 b^2 d^{3/2}}+\frac{x^2 \sqrt{c+d x^4}}{4 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^9/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

(x^2*Sqrt[c + d*x^4])/(4*b*d) + (a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*S
qrt[c + d*x^4])])/(2*b^2*Sqrt[b*c - a*d]) - ((b*c + 2*a*d)*ArcTanh[(Sqrt[d]*x^2)
/Sqrt[c + d*x^4]])/(4*b^2*d^(3/2))

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Rubi in Sympy [A]  time = 45.6418, size = 107, normalized size = 0.87 \[ \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 b^{2} \sqrt{a d - b c}} + \frac{x^{2} \sqrt{c + d x^{4}}}{4 b d} - \frac{\left (2 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c + d x^{4}}} \right )}}{4 b^{2} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

a**(3/2)*atanh(x**2*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**4)))/(2*b**2*sqrt(a*d
- b*c)) + x**2*sqrt(c + d*x**4)/(4*b*d) - (2*a*d + b*c)*atanh(sqrt(d)*x**2/sqrt
(c + d*x**4))/(4*b**2*d**(3/2))

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Mathematica [A]  time = 0.217319, size = 118, normalized size = 0.96 \[ \frac{\frac{2 a^{3/2} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{\sqrt{b c-a d}}-\frac{(2 a d+b c) \log \left (\sqrt{d} \sqrt{c+d x^4}+d x^2\right )}{d^{3/2}}+\frac{b x^2 \sqrt{c+d x^4}}{d}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

((b*x^2*Sqrt[c + d*x^4])/d + (2*a^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sq
rt[c + d*x^4])])/Sqrt[b*c - a*d] - ((b*c + 2*a*d)*Log[d*x^2 + Sqrt[d]*Sqrt[c + d
*x^4]])/d^(3/2))/(4*b^2)

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Maple [B]  time = 0.037, size = 408, normalized size = 3.3 \[{\frac{{x}^{2}}{4\,bd}\sqrt{d{x}^{4}+c}}-{\frac{c}{4\,b}\ln \left ({x}^{2}\sqrt{d}+\sqrt{d{x}^{4}+c} \right ){d}^{-{\frac{3}{2}}}}-{\frac{a}{2\,{b}^{2}}\ln \left ({x}^{2}\sqrt{d}+\sqrt{d{x}^{4}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{4\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{{a}^{2}}{4\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

1/4*x^2*(d*x^4+c)^(1/2)/b/d-1/4/b*c/d^(3/2)*ln(x^2*d^(1/2)+(d*x^4+c)^(1/2))-1/2/
b^2*a*ln(x^2*d^(1/2)+(d*x^4+c)^(1/2))/d^(1/2)-1/4*a^2/b^2/(-a*b)^(1/2)/(-(a*d-b*
c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))+2*(-(a*
d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(
1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*(-a*b)^(1/2)))+1/4*a^2/b^2/(-a*b)^(1/2)/(-(a*
d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))+2*(
-(a*d-b*c)/b)^(1/2)*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*
b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2+1/b*(-a*b)^(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(x^9/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.362422, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(d*x^4 + c)*b*sqrt(d)*x^2 + a*d^(3/2)*sqrt(-a/(b*c - a*d))*log(((b^2
*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 + 4*
((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^6 - (a*b*c^2 - a^2*c*d)*x^2)*sqrt(d*x^4 + c
)*sqrt(-a/(b*c - a*d)))/(b^2*x^8 + 2*a*b*x^4 + a^2)) + (b*c + 2*a*d)*log(2*sqrt(
d*x^4 + c)*d*x^2 - (2*d*x^4 + c)*sqrt(d)))/(b^2*d^(3/2)), 1/8*(2*sqrt(d*x^4 + c)
*b*sqrt(-d)*x^2 + a*sqrt(-d)*d*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d +
8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a^2*c^2 + 4*((b^2*c^2 - 3*a*b*c
*d + 2*a^2*d^2)*x^6 - (a*b*c^2 - a^2*c*d)*x^2)*sqrt(d*x^4 + c)*sqrt(-a/(b*c - a*
d)))/(b^2*x^8 + 2*a*b*x^4 + a^2)) - 2*(b*c + 2*a*d)*arctan(sqrt(-d)*x^2/sqrt(d*x
^4 + c)))/(b^2*sqrt(-d)*d), 1/8*(2*sqrt(d*x^4 + c)*b*sqrt(d)*x^2 + 2*a*d^(3/2)*s
qrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)*(b*c -
a*d)*x^2*sqrt(a/(b*c - a*d)))) + (b*c + 2*a*d)*log(2*sqrt(d*x^4 + c)*d*x^2 - (2*
d*x^4 + c)*sqrt(d)))/(b^2*d^(3/2)), 1/4*(sqrt(d*x^4 + c)*b*sqrt(-d)*x^2 + a*sqrt
(-d)*d*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 + c)
*(b*c - a*d)*x^2*sqrt(a/(b*c - a*d)))) - (b*c + 2*a*d)*arctan(sqrt(-d)*x^2/sqrt(
d*x^4 + c)))/(b^2*sqrt(-d)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral(x**9/((a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.292656, size = 140, normalized size = 1.14 \[ \frac{\sqrt{d x^{4} + c} x^{2}}{4 \, b d} - \frac{a^{2} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{2 \, \sqrt{a b c - a^{2} d} b^{2}} + \frac{{\left (b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{d + \frac{c}{x^{4}}}}{\sqrt{-d}}\right )}{4 \, b^{2} \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

1/4*sqrt(d*x^4 + c)*x^2/(b*d) - 1/2*a^2*arctan(a*sqrt(d + c/x^4)/sqrt(a*b*c - a^
2*d))/(sqrt(a*b*c - a^2*d)*b^2) + 1/4*(b*c + 2*a*d)*arctan(sqrt(d + c/x^4)/sqrt(
-d))/(b^2*sqrt(-d)*d)