3.655 \(\int \frac{x^7}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=99 \[ \frac{a \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.228526, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{a \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.071, size = 80, normalized size = 0.81 \[ - \frac{a \sqrt{c + d x^{4}}}{4 b \left (a + b x^{4}\right ) \left (a d - b c\right )} + \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.125253, size = 99, normalized size = 1. \[ \frac{a \sqrt{c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

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

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Maple [B]  time = 0.016, size = 851, normalized size = 8.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

-1/4/b^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/b^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)))+1/8/b^2*(-a*b)^(1/2)/(a*d
-b*c)/(x^2-1/b*(-a*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)+1/8*a/b^2*d/(a*d-b*c)/(-(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)))-1/8/b^2*(-a*b)^(1/2)/(a*d-b*c)/(x^2+1/b*(
-a*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)+1/8*a/b^2*d/(a*d-b*c)/(-(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^7/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243263, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d} a +{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + 2 \, a b c - a^{2} d\right )} \log \left (\frac{{\left (b d x^{4} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{b x^{4} + a}\right )}{8 \,{\left ({\left (b^{3} c - a b^{2} d\right )} x^{4} + a b^{2} c - a^{2} b d\right )} \sqrt{b^{2} c - a b d}}, \frac{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d} a -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} + 2 \, a b c - a^{2} d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left ({\left (b^{3} c - a b^{2} d\right )} x^{4} + a b^{2} c - a^{2} b d\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.215374, size = 157, normalized size = 1.59 \[ \frac{\frac{\sqrt{d x^{4} + c} a d^{2}}{{\left (b^{2} c - a b d\right )}{\left ({\left (d x^{4} + c\right )} b - b c + a d\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

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