3.347 \(\int \frac{x^{7/2}}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=251 \[ \frac{77 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}+\frac{77 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{15/4}}-\frac{77}{48 b^3 x^{3/2}}+\frac{11}{16 b^2 x^{3/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{3/2} \left (b+c x^2\right )^2} \]

[Out]

-77/(48*b^3*x^(3/2)) + 1/(4*b*x^(3/2)*(b + c*x^2)^2) + 11/(16*b^2*x^(3/2)*(b + c
*x^2)) + (77*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*
b^(15/4)) - (77*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[
2]*b^(15/4)) + (77*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[
c]*x])/(64*Sqrt[2]*b^(15/4)) - (77*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(15/4))

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Rubi [A]  time = 0.430924, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{77 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{15/4}}+\frac{77 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{15/4}}-\frac{77 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{15/4}}-\frac{77}{48 b^3 x^{3/2}}+\frac{11}{16 b^2 x^{3/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{3/2} \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(b*x^2 + c*x^4)^3,x]

[Out]

-77/(48*b^3*x^(3/2)) + 1/(4*b*x^(3/2)*(b + c*x^2)^2) + 11/(16*b^2*x^(3/2)*(b + c
*x^2)) + (77*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*
b^(15/4)) - (77*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[
2]*b^(15/4)) + (77*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[
c]*x])/(64*Sqrt[2]*b^(15/4)) - (77*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(15/4))

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Rubi in Sympy [A]  time = 81.4421, size = 238, normalized size = 0.95 \[ \frac{1}{4 b x^{\frac{3}{2}} \left (b + c x^{2}\right )^{2}} + \frac{11}{16 b^{2} x^{\frac{3}{2}} \left (b + c x^{2}\right )} - \frac{77}{48 b^{3} x^{\frac{3}{2}}} + \frac{77 \sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{15}{4}}} - \frac{77 \sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{15}{4}}} + \frac{77 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{15}{4}}} - \frac{77 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{15}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(c*x**4+b*x**2)**3,x)

[Out]

1/(4*b*x**(3/2)*(b + c*x**2)**2) + 11/(16*b**2*x**(3/2)*(b + c*x**2)) - 77/(48*b
**3*x**(3/2)) + 77*sqrt(2)*c**(3/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqr
t(b) + sqrt(c)*x)/(128*b**(15/4)) - 77*sqrt(2)*c**(3/4)*log(sqrt(2)*b**(1/4)*c**
(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(15/4)) + 77*sqrt(2)*c**(3/4)*atan(
1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(15/4)) - 77*sqrt(2)*c**(3/4)*atan
(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(15/4))

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Mathematica [A]  time = 0.209879, size = 234, normalized size = 0.93 \[ \frac{-\frac{96 b^{7/4} c \sqrt{x}}{\left (b+c x^2\right )^2}-\frac{360 b^{3/4} c \sqrt{x}}{b+c x^2}-\frac{256 b^{3/4}}{x^{3/2}}+231 \sqrt{2} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-231 \sqrt{2} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+462 \sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-462 \sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{384 b^{15/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(b*x^2 + c*x^4)^3,x]

[Out]

((-256*b^(3/4))/x^(3/2) - (96*b^(7/4)*c*Sqrt[x])/(b + c*x^2)^2 - (360*b^(3/4)*c*
Sqrt[x])/(b + c*x^2) + 462*Sqrt[2]*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/
b^(1/4)] - 462*Sqrt[2]*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 2
31*Sqrt[2]*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] -
231*Sqrt[2]*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/
(384*b^(15/4))

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Maple [A]  time = 0.025, size = 181, normalized size = 0.7 \[ -{\frac{15\,{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{19\,c}{16\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{77\,c\sqrt{2}}{128\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{77\,c\sqrt{2}}{64\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{77\,c\sqrt{2}}{64\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{2}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(c*x^4+b*x^2)^3,x)

[Out]

-15/16/b^3*c^2/(c*x^2+b)^2*x^(5/2)-19/16/b^2*c/(c*x^2+b)^2*x^(1/2)-77/128/b^4*c*
(b/c)^(1/4)*2^(1/2)*ln((x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x-(b/c)^(1/4
)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))-77/64/b^4*c*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
b/c)^(1/4)*x^(1/2)+1)-77/64/b^4*c*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)
*x^(1/2)-1)-2/3/b^3/x^(3/2)

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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^(7/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290843, size = 356, normalized size = 1.42 \[ -\frac{308 \, c^{2} x^{4} + 484 \, b c x^{2} - 924 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}}}{c \sqrt{x} + \sqrt{b^{8} \sqrt{-\frac{c^{3}}{b^{15}}} + c^{2} x}}\right ) + 231 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} \log \left (77 \, b^{4} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} + 77 \, c \sqrt{x}\right ) - 231 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} \log \left (-77 \, b^{4} \left (-\frac{c^{3}}{b^{15}}\right )^{\frac{1}{4}} + 77 \, c \sqrt{x}\right ) + 128 \, b^{2}}{192 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

-1/192*(308*c^2*x^4 + 484*b*c*x^2 - 924*(b^3*c^2*x^5 + 2*b^4*c*x^3 + b^5*x)*sqrt
(x)*(-c^3/b^15)^(1/4)*arctan(b^4*(-c^3/b^15)^(1/4)/(c*sqrt(x) + sqrt(b^8*sqrt(-c
^3/b^15) + c^2*x))) + 231*(b^3*c^2*x^5 + 2*b^4*c*x^3 + b^5*x)*sqrt(x)*(-c^3/b^15
)^(1/4)*log(77*b^4*(-c^3/b^15)^(1/4) + 77*c*sqrt(x)) - 231*(b^3*c^2*x^5 + 2*b^4*
c*x^3 + b^5*x)*sqrt(x)*(-c^3/b^15)^(1/4)*log(-77*b^4*(-c^3/b^15)^(1/4) + 77*c*sq
rt(x)) + 128*b^2)/((b^3*c^2*x^5 + 2*b^4*c*x^3 + b^5*x)*sqrt(x))

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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**(7/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280037, size = 281, normalized size = 1.12 \[ -\frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{4}} - \frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{4}} - \frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{4}} + \frac{77 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{4}} - \frac{15 \, c^{2} x^{\frac{5}{2}} + 19 \, b c \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{3}} - \frac{2}{3 \, b^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

-77/64*sqrt(2)*(b*c^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x)
)/(b/c)^(1/4))/b^4 - 77/64*sqrt(2)*(b*c^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b
/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/b^4 - 77/128*sqrt(2)*(b*c^3)^(1/4)*ln(sqrt(2
)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 + 77/128*sqrt(2)*(b*c^3)^(1/4)*ln(-sq
rt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 - 1/16*(15*c^2*x^(5/2) + 19*b*c*s
qrt(x))/((c*x^2 + b)^2*b^3) - 2/3/(b^3*x^(3/2))