3.192 \(\int \frac{A+B x^2}{x^{5/2} \left (b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=257 \[ \frac{c^{3/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}-\frac{c^{3/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{c^{3/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{11/4}}-\frac{c^{3/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{11/4}}-\frac{2 (b B-A c)}{3 b^2 x^{3/2}}-\frac{2 A}{7 b x^{7/2}} \]

[Out]

(-2*A)/(7*b*x^(7/2)) - (2*(b*B - A*c))/(3*b^2*x^(3/2)) + (c^(3/4)*(b*B - A*c)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(11/4)) - (c^(3/4)*(b*B
- A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(11/4)) + (c^(3
/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*S
qrt[2]*b^(11/4)) - (c^(3/4)*(b*B - A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(11/4))

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Rubi [A]  time = 0.449216, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{c^{3/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}-\frac{c^{3/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{c^{3/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{11/4}}-\frac{c^{3/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{11/4}}-\frac{2 (b B-A c)}{3 b^2 x^{3/2}}-\frac{2 A}{7 b x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(5/2)*(b*x^2 + c*x^4)),x]

[Out]

(-2*A)/(7*b*x^(7/2)) - (2*(b*B - A*c))/(3*b^2*x^(3/2)) + (c^(3/4)*(b*B - A*c)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(11/4)) - (c^(3/4)*(b*B
- A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(11/4)) + (c^(3
/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*S
qrt[2]*b^(11/4)) - (c^(3/4)*(b*B - A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(11/4))

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Rubi in Sympy [A]  time = 71.5566, size = 241, normalized size = 0.94 \[ - \frac{2 A}{7 b x^{\frac{7}{2}}} + \frac{2 \left (A c - B b\right )}{3 b^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (A c - B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (A c - B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{11}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(5/2)/(c*x**4+b*x**2),x)

[Out]

-2*A/(7*b*x**(7/2)) + 2*(A*c - B*b)/(3*b**2*x**(3/2)) - sqrt(2)*c**(3/4)*(A*c -
B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(11/4))
+ sqrt(2)*c**(3/4)*(A*c - B*b)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b)
+ sqrt(c)*x)/(4*b**(11/4)) - sqrt(2)*c**(3/4)*(A*c - B*b)*atan(1 - sqrt(2)*c**(1
/4)*sqrt(x)/b**(1/4))/(2*b**(11/4)) + sqrt(2)*c**(3/4)*(A*c - B*b)*atan(1 + sqrt
(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(11/4))

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Mathematica [A]  time = 0.361466, size = 243, normalized size = 0.95 \[ \frac{\frac{56 b^{3/4} (A c-b B)}{x^{3/2}}-\frac{24 A b^{7/4}}{x^{7/2}}+21 \sqrt{2} c^{3/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+21 \sqrt{2} c^{3/4} (A c-b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-42 \sqrt{2} c^{3/4} (A c-b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+42 \sqrt{2} c^{3/4} (A c-b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{84 b^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(5/2)*(b*x^2 + c*x^4)),x]

[Out]

((-24*A*b^(7/4))/x^(7/2) + (56*b^(3/4)*(-(b*B) + A*c))/x^(3/2) - 42*Sqrt[2]*c^(3
/4)*(-(b*B) + A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 42*Sqrt[2]*c^
(3/4)*(-(b*B) + A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 21*Sqrt[2]*
c^(3/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] +
21*Sqrt[2]*c^(3/4)*(-(b*B) + A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x]
+ Sqrt[c]*x])/(84*b^(11/4))

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Maple [A]  time = 0.017, size = 308, normalized size = 1.2 \[ -{\frac{2\,A}{7\,b}{x}^{-{\frac{7}{2}}}}+{\frac{2\,Ac}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,b}{x}^{-{\frac{3}{2}}}}+{\frac{{c}^{2}\sqrt{2}A}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{{c}^{2}\sqrt{2}A}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{{c}^{2}\sqrt{2}A}{4\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{c\sqrt{2}B}{2\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{c\sqrt{2}B}{2\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{c\sqrt{2}B}{4\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(5/2)/(c*x^4+b*x^2),x)

[Out]

-2/7*A/b/x^(7/2)+2/3/x^(3/2)/b^2*A*c-2/3/x^(3/2)/b*B+1/2*c^2/b^3*(b/c)^(1/4)*2^(
1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+1/2*c^2/b^3*(b/c)^(1/4)*2^(1/2)*A*a
rctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+1/4*c^2/b^3*(b/c)^(1/4)*2^(1/2)*A*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)
))-1/2*c/b^2*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-1/2*c/b
^2*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-1/4*c/b^2*(b/c)^(
1/4)*2^(1/2)*B*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)))

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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((B*x^2 + A)/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270958, size = 787, normalized size = 3.06 \[ -\frac{84 \, b^{2} x^{\frac{7}{2}} \left (-\frac{B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{3} \left (-\frac{B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac{1}{4}}}{{\left (B b c - A c^{2}\right )} \sqrt{x} - \sqrt{b^{6} \sqrt{-\frac{B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}} +{\left (B^{2} b^{2} c^{2} - 2 \, A B b c^{3} + A^{2} c^{4}\right )} x}}\right ) - 21 \, b^{2} x^{\frac{7}{2}} \left (-\frac{B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac{1}{4}} \log \left (b^{3} \left (-\frac{B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac{1}{4}} -{\left (B b c - A c^{2}\right )} \sqrt{x}\right ) + 21 \, b^{2} x^{\frac{7}{2}} \left (-\frac{B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-b^{3} \left (-\frac{B^{4} b^{4} c^{3} - 4 \, A B^{3} b^{3} c^{4} + 6 \, A^{2} B^{2} b^{2} c^{5} - 4 \, A^{3} B b c^{6} + A^{4} c^{7}}{b^{11}}\right )^{\frac{1}{4}} -{\left (B b c - A c^{2}\right )} \sqrt{x}\right ) + 28 \,{\left (B b - A c\right )} x^{2} + 12 \, A b}{42 \, b^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="fricas")

[Out]

-1/42*(84*b^2*x^(7/2)*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A
^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4)*arctan(-b^3*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 +
6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4)/((B*b*c - A*c^2)*sqrt(
x) - sqrt(b^6*sqrt(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B
*b*c^6 + A^4*c^7)/b^11) + (B^2*b^2*c^2 - 2*A*B*b*c^3 + A^2*c^4)*x))) - 21*b^2*x^
(7/2)*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4
*c^7)/b^11)^(1/4)*log(b^3*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 -
4*A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4) - (B*b*c - A*c^2)*sqrt(x)) + 21*b^2*x^(7/2
)*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*A^3*B*b*c^6 + A^4*c^7
)/b^11)^(1/4)*log(-b^3*(-(B^4*b^4*c^3 - 4*A*B^3*b^3*c^4 + 6*A^2*B^2*b^2*c^5 - 4*
A^3*B*b*c^6 + A^4*c^7)/b^11)^(1/4) - (B*b*c - A*c^2)*sqrt(x)) + 28*(B*b - A*c)*x
^2 + 12*A*b)/(b^2*x^(7/2))

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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((B*x**2+A)/x**(5/2)/(c*x**4+b*x**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219168, size = 347, normalized size = 1.35 \[ -\frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b - \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3}} - \frac{2 \,{\left (7 \, B b x^{2} - 7 \, A c x^{2} + 3 \, A b\right )}}{21 \, b^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((b*c^3)^(1/4)*B*b - (b*c^3)^(1/4)*A*c)*arctan(1/2*sqrt(2)*(sqrt(2)
*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/b^3 - 1/2*sqrt(2)*((b*c^3)^(1/4)*B*b - (b
*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/
4))/b^3 - 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b - (b*c^3)^(1/4)*A*c)*ln(sqrt(2)*sqrt(x)
*(b/c)^(1/4) + x + sqrt(b/c))/b^3 + 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b - (b*c^3)^(1/
4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^3 - 2/21*(7*B*b*x^2 -
7*A*c*x^2 + 3*A*b)/(b^2*x^(7/2))