3.741 \(\int \frac{x^7}{\left (a+b x^8\right )^2 \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=87 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \]

[Out]

-Sqrt[c + d*x^8]/(8*(b*c - a*d)*(a + b*x^8)) + (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^
8])/Sqrt[b*c - a*d]])/(8*Sqrt[b]*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.208327, antiderivative size = 87, 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 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^8}}{8 \left (a+b x^8\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^7/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-Sqrt[c + d*x^8]/(8*(b*c - a*d)*(a + b*x^8)) + (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^
8])/Sqrt[b*c - a*d]])/(8*Sqrt[b]*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 21.1248, size = 70, normalized size = 0.8 \[ \frac{\sqrt{c + d x^{8}}}{8 \left (a + b x^{8}\right ) \left (a d - b c\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{8 \sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

sqrt(c + d*x**8)/(8*(a + b*x**8)*(a*d - b*c)) + d*atan(sqrt(b)*sqrt(c + d*x**8)/
sqrt(a*d - b*c))/(8*sqrt(b)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.142937, size = 84, normalized size = 0.97 \[ \frac{\frac{\sqrt{c+d x^8}}{a+b x^8}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{8 a d-8 b c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

(Sqrt[c + d*x^8]/(a + b*x^8) - (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a
*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/(-8*b*c + 8*a*d)

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{{x}^{7}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(x^7/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

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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^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231832, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b d x^{8} + a d\right )} \log \left (\frac{{\left (b d x^{8} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{8} + c}{\left (b^{2} c - a b d\right )}}{b x^{8} + a}\right ) + 2 \, \sqrt{d x^{8} + c} \sqrt{b^{2} c - a b d}}{16 \,{\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (b d x^{8} + a d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}\right ) - \sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}{8 \,{\left ({\left (b^{2} c - a b d\right )} x^{8} + a b c - a^{2} d\right )} \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

[-1/16*((b*d*x^8 + a*d)*log(((b*d*x^8 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d) - 2*sqr
t(d*x^8 + c)*(b^2*c - a*b*d))/(b*x^8 + a)) + 2*sqrt(d*x^8 + c)*sqrt(b^2*c - a*b*
d))/(((b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*sqrt(b^2*c - a*b*d)), 1/8*((b*d*x^8 +
a*d)*arctan(-(b*c - a*d)/(sqrt(d*x^8 + c)*sqrt(-b^2*c + a*b*d))) - sqrt(d*x^8 +
c)*sqrt(-b^2*c + a*b*d))/(((b^2*c - a*b*d)*x^8 + a*b*c - a^2*d)*sqrt(-b^2*c + a
*b*d))]

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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**7/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213325, size = 124, normalized size = 1.43 \[ -\frac{1}{8} \, d{\left (\frac{\arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{\sqrt{d x^{8} + c}}{{\left ({\left (d x^{8} + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

-1/8*d*(arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*(b*
c - a*d)) + sqrt(d*x^8 + c)/(((d*x^8 + c)*b - b*c + a*d)*(b*c - a*d)))