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

Optimal. Leaf size=230 \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4}}-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )} \]

[Out]

-7/(6*b^2*x^(3/2)) + 1/(2*b*x^(3/2)*(b + c*x^2)) + (7*c^(3/4)*ArcTan[1 - (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(11/4)) - (7*c^(3/4)*ArcTan[1 + (Sqrt[
2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(11/4)) + (7*c^(3/4)*Log[Sqrt[b] - Sq
rt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4)) - (7*c^(3/4)*Lo
g[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4))

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Rubi [A]  time = 0.378972, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{7 c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{11/4}}+\frac{7 c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{11/4}}-\frac{7 c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{11/4}}-\frac{7}{6 b^2 x^{3/2}}+\frac{1}{2 b x^{3/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-7/(6*b^2*x^(3/2)) + 1/(2*b*x^(3/2)*(b + c*x^2)) + (7*c^(3/4)*ArcTan[1 - (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(11/4)) - (7*c^(3/4)*ArcTan[1 + (Sqrt[
2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(11/4)) + (7*c^(3/4)*Log[Sqrt[b] - Sq
rt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4)) - (7*c^(3/4)*Lo
g[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(11/4))

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Rubi in Sympy [A]  time = 70.957, size = 218, normalized size = 0.95 \[ \frac{1}{2 b x^{\frac{3}{2}} \left (b + c x^{2}\right )} - \frac{7}{6 b^{2} x^{\frac{3}{2}}} + \frac{7 \sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{11}{4}}} - \frac{7 \sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{11}{4}}} + \frac{7 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{11}{4}}} - \frac{7 \sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*b*x**(3/2)*(b + c*x**2)) - 7/(6*b**2*x**(3/2)) + 7*sqrt(2)*c**(3/4)*log(-sq
rt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(11/4)) - 7*sqrt(2
)*c**(3/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(
11/4)) + 7*sqrt(2)*c**(3/4)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(1
1/4)) - 7*sqrt(2)*c**(3/4)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(11
/4))

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Mathematica [A]  time = 0.382629, size = 212, normalized size = 0.92 \[ \frac{-\frac{24 b^{3/4} c \sqrt{x}}{b+c x^2}-\frac{32 b^{3/4}}{x^{3/2}}+21 \sqrt{2} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-21 \sqrt{2} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+42 \sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-42 \sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{48 b^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*b^(3/4))/x^(3/2) - (24*b^(3/4)*c*Sqrt[x])/(b + c*x^2) + 42*Sqrt[2]*c^(3/4)
*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 42*Sqrt[2]*c^(3/4)*ArcTan[1 + (
Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 21*Sqrt[2]*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(
1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 21*Sqrt[2]*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(
1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(48*b^(11/4))

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Maple [A]  time = 0.019, size = 161, normalized size = 0.7 \[ -{\frac{c}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{7\,c\sqrt{2}}{16\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{7\,c\sqrt{2}}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{7\,c\sqrt{2}}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{2}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(c*x^4+b*x^2)^2,x)

[Out]

-1/2/b^2*c*x^(1/2)/(c*x^2+b)-7/16/b^3*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)))-7/8/b^3*
c*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-7/8/b^3*c*(b/c)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-2/3/b^2/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^(3/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286773, size = 282, normalized size = 1.23 \[ -\frac{28 \, c x^{2} - 84 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{3} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}}}{c \sqrt{x} + \sqrt{b^{6} \sqrt{-\frac{c^{3}}{b^{11}}} + c^{2} x}}\right ) + 21 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} \log \left (7 \, b^{3} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} + 7 \, c \sqrt{x}\right ) - 21 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-7 \, b^{3} \left (-\frac{c^{3}}{b^{11}}\right )^{\frac{1}{4}} + 7 \, c \sqrt{x}\right ) + 16 \, b}{24 \,{\left (b^{2} c x^{3} + b^{3} x\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(28*c*x^2 - 84*(b^2*c*x^3 + b^3*x)*sqrt(x)*(-c^3/b^11)^(1/4)*arctan(b^3*(-
c^3/b^11)^(1/4)/(c*sqrt(x) + sqrt(b^6*sqrt(-c^3/b^11) + c^2*x))) + 21*(b^2*c*x^3
+ b^3*x)*sqrt(x)*(-c^3/b^11)^(1/4)*log(7*b^3*(-c^3/b^11)^(1/4) + 7*c*sqrt(x)) -
21*(b^2*c*x^3 + b^3*x)*sqrt(x)*(-c^3/b^11)^(1/4)*log(-7*b^3*(-c^3/b^11)^(1/4) +
7*c*sqrt(x)) + 16*b)/((b^2*c*x^3 + b^3*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**(3/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280238, size = 265, normalized size = 1.15 \[ -\frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{3}} + \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{3}} - \frac{c \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} b^{2}} - \frac{2}{3 \, b^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/8*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^3 - 7/8*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^3 - 7/16*sqrt(2)*(b*c^3)^(1/4)*ln(sqrt(2)*sqrt
(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^3 + 7/16*sqrt(2)*(b*c^3)^(1/4)*ln(-sqrt(2)*sq
rt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^3 - 1/2*c*sqrt(x)/((c*x^2 + b)*b^2) - 2/3/(
b^2*x^(3/2))