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

Optimal. Leaf size=276 \[ \frac{c^{5/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{c^{5/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{c^{5/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{13/4}}+\frac{c^{5/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{13/4}}+\frac{2 c (b B-A c)}{b^3 \sqrt{x}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{9 b x^{9/2}} \]

[Out]

(-2*A)/(9*b*x^(9/2)) - (2*(b*B - A*c))/(5*b^2*x^(5/2)) + (2*c*(b*B - A*c))/(b^3*
Sqrt[x]) - (c^(5/4)*(b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(
Sqrt[2]*b^(13/4)) + (c^(5/4)*(b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^
(1/4)])/(Sqrt[2]*b^(13/4)) + (c^(5/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*
c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4)) - (c^(5/4)*(b*B - A*c)*Log[Sq
rt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4))

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Rubi [A]  time = 0.503335, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{c^{5/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{c^{5/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}-\frac{c^{5/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{13/4}}+\frac{c^{5/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{13/4}}+\frac{2 c (b B-A c)}{b^3 \sqrt{x}}-\frac{2 (b B-A c)}{5 b^2 x^{5/2}}-\frac{2 A}{9 b x^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(9*b*x^(9/2)) - (2*(b*B - A*c))/(5*b^2*x^(5/2)) + (2*c*(b*B - A*c))/(b^3*
Sqrt[x]) - (c^(5/4)*(b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(
Sqrt[2]*b^(13/4)) + (c^(5/4)*(b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^
(1/4)])/(Sqrt[2]*b^(13/4)) + (c^(5/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*
c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4)) - (c^(5/4)*(b*B - A*c)*Log[Sq
rt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4))

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Rubi in Sympy [A]  time = 81.3397, size = 260, normalized size = 0.94 \[ - \frac{2 A}{9 b x^{\frac{9}{2}}} + \frac{2 \left (A c - B b\right )}{5 b^{2} x^{\frac{5}{2}}} - \frac{2 c \left (A c - B b\right )}{b^{3} \sqrt{x}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (A c - B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{13}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (A c - B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{13}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{13}{4}}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(9*b*x**(9/2)) + 2*(A*c - B*b)/(5*b**2*x**(5/2)) - 2*c*(A*c - B*b)/(b**3*sq
rt(x)) - sqrt(2)*c**(5/4)*(A*c - B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + s
qrt(b) + sqrt(c)*x)/(4*b**(13/4)) + sqrt(2)*c**(5/4)*(A*c - B*b)*log(sqrt(2)*b**
(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(13/4)) + sqrt(2)*c**(5/4)*(
A*c - B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(13/4)) - sqrt(2)*c
**(5/4)*(A*c - B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(13/4))

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Mathematica [A]  time = 0.519103, size = 264, normalized size = 0.96 \[ \frac{\frac{72 b^{5/4} (A c-b B)}{x^{5/2}}-\frac{40 A b^{9/4}}{x^{9/2}}+45 \sqrt{2} c^{5/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} c^{5/4} (A c-b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+90 \sqrt{2} c^{5/4} (A c-b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} c^{5/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{360 \sqrt [4]{b} c (b B-A c)}{\sqrt{x}}}{180 b^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-40*A*b^(9/4))/x^(9/2) + (72*b^(5/4)*(-(b*B) + A*c))/x^(5/2) + (360*b^(1/4)*c*
(b*B - A*c))/Sqrt[x] + 90*Sqrt[2]*c^(5/4)*(-(b*B) + A*c)*ArcTan[1 - (Sqrt[2]*c^(
1/4)*Sqrt[x])/b^(1/4)] + 90*Sqrt[2]*c^(5/4)*(b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1
/4)*Sqrt[x])/b^(1/4)] + 45*Sqrt[2]*c^(5/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(
1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*c^(5/4)*(-(b*B) + A*c)*Log[Sqrt[b
] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(180*b^(13/4))

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Maple [A]  time = 0.019, size = 330, normalized size = 1.2 \[ -{\frac{2\,A}{9\,b}{x}^{-{\frac{9}{2}}}}+{\frac{2\,Ac}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,b}{x}^{-{\frac{5}{2}}}}-2\,{\frac{A{c}^{2}}{{b}^{3}\sqrt{x}}}+2\,{\frac{Bc}{{b}^{2}\sqrt{x}}}-{\frac{{c}^{2}\sqrt{2}A}{2\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{{c}^{2}\sqrt{2}A}{4\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{{c}^{2}\sqrt{2}A}{2\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{c\sqrt{2}B}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{c\sqrt{2}B}{4\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{c\sqrt{2}B}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(7/2)/(c*x^4+b*x^2),x)

[Out]

-2/9*A/b/x^(9/2)+2/5/x^(5/2)/b^2*A*c-2/5/x^(5/2)/b*B-2/b^3*c^2/x^(1/2)*A+2/b^2*c
/x^(1/2)*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)-1/4*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^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/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)))+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)

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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^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25254, size = 1089, normalized size = 3.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90*(180*b^3*x^(9/2)*(-(B^4*b^4*c^5 - 4*A*B^3*b^3*c^6 + 6*A^2*B^2*b^2*c^7 - 4*
A^3*B*b*c^8 + A^4*c^9)/b^13)^(1/4)*arctan(-b^10*(-(B^4*b^4*c^5 - 4*A*B^3*b^3*c^6
+ 6*A^2*B^2*b^2*c^7 - 4*A^3*B*b*c^8 + A^4*c^9)/b^13)^(3/4)/((B^3*b^3*c^4 - 3*A*
B^2*b^2*c^5 + 3*A^2*B*b*c^6 - A^3*c^7)*sqrt(x) - sqrt((B^6*b^6*c^8 - 6*A*B^5*b^5
*c^9 + 15*A^2*B^4*b^4*c^10 - 20*A^3*B^3*b^3*c^11 + 15*A^4*B^2*b^2*c^12 - 6*A^5*B
*b*c^13 + A^6*c^14)*x - (B^4*b^11*c^5 - 4*A*B^3*b^10*c^6 + 6*A^2*B^2*b^9*c^7 - 4
*A^3*B*b^8*c^8 + A^4*b^7*c^9)*sqrt(-(B^4*b^4*c^5 - 4*A*B^3*b^3*c^6 + 6*A^2*B^2*b
^2*c^7 - 4*A^3*B*b*c^8 + A^4*c^9)/b^13)))) + 45*b^3*x^(9/2)*(-(B^4*b^4*c^5 - 4*A
*B^3*b^3*c^6 + 6*A^2*B^2*b^2*c^7 - 4*A^3*B*b*c^8 + A^4*c^9)/b^13)^(1/4)*log(b^10
*(-(B^4*b^4*c^5 - 4*A*B^3*b^3*c^6 + 6*A^2*B^2*b^2*c^7 - 4*A^3*B*b*c^8 + A^4*c^9)
/b^13)^(3/4) - (B^3*b^3*c^4 - 3*A*B^2*b^2*c^5 + 3*A^2*B*b*c^6 - A^3*c^7)*sqrt(x)
) - 45*b^3*x^(9/2)*(-(B^4*b^4*c^5 - 4*A*B^3*b^3*c^6 + 6*A^2*B^2*b^2*c^7 - 4*A^3*
B*b*c^8 + A^4*c^9)/b^13)^(1/4)*log(-b^10*(-(B^4*b^4*c^5 - 4*A*B^3*b^3*c^6 + 6*A^
2*B^2*b^2*c^7 - 4*A^3*B*b*c^8 + A^4*c^9)/b^13)^(3/4) - (B^3*b^3*c^4 - 3*A*B^2*b^
2*c^5 + 3*A^2*B*b*c^6 - A^3*c^7)*sqrt(x)) - 180*(B*b*c - A*c^2)*x^4 + 20*A*b^2 +
36*(B*b^2 - A*b*c)*x^2)/(b^3*x^(9/2))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224691, size = 393, normalized size = 1.42 \[ \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4} c} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4} c} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4} c} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4} c} + \frac{2 \,{\left (45 \, B b c x^{4} - 45 \, A c^{2} x^{4} - 9 \, B b^{2} x^{2} + 9 \, A b c x^{2} - 5 \, A b^{2}\right )}}{45 \, b^{3} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*((b*c^3)^(3/4)*B*b - (b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(sqrt(2)*
(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^4*c) + 1/2*sqrt(2)*((b*c^3)^(3/4)*B*b -
(b*c^3)^(3/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^
(1/4))/(b^4*c) - 1/4*sqrt(2)*((b*c^3)^(3/4)*B*b - (b*c^3)^(3/4)*A*c)*ln(sqrt(2)*
sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c) + 1/4*sqrt(2)*((b*c^3)^(3/4)*B*b -
(b*c^3)^(3/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c) + 2/
45*(45*B*b*c*x^4 - 45*A*c^2*x^4 - 9*B*b^2*x^2 + 9*A*b*c*x^2 - 5*A*b^2)/(b^3*x^(9
/2))