3.126 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{c^2 (6 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac{\left (b x^2+c x^4\right )^{3/2} (6 b B-A c)}{24 b x^7}-\frac{c \sqrt{b x^2+c x^4} (6 b B-A c)}{16 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}} \]

[Out]

-(c*(6*b*B - A*c)*Sqrt[b*x^2 + c*x^4])/(16*b*x^3) - ((6*b*B - A*c)*(b*x^2 + c*x^
4)^(3/2))/(24*b*x^7) - (A*(b*x^2 + c*x^4)^(5/2))/(6*b*x^11) - (c^2*(6*b*B - A*c)
*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(16*b^(3/2))

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Rubi [A]  time = 0.36742, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{c^2 (6 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac{\left (b x^2+c x^4\right )^{3/2} (6 b B-A c)}{24 b x^7}-\frac{c \sqrt{b x^2+c x^4} (6 b B-A c)}{16 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^10,x]

[Out]

-(c*(6*b*B - A*c)*Sqrt[b*x^2 + c*x^4])/(16*b*x^3) - ((6*b*B - A*c)*(b*x^2 + c*x^
4)^(3/2))/(24*b*x^7) - (A*(b*x^2 + c*x^4)^(5/2))/(6*b*x^11) - (c^2*(6*b*B - A*c)
*ArcTanh[(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(16*b^(3/2))

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Rubi in Sympy [A]  time = 28.8092, size = 121, normalized size = 0.86 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{6 b x^{11}} + \frac{c \left (A c - 6 B b\right ) \sqrt{b x^{2} + c x^{4}}}{16 b x^{3}} + \frac{\left (A c - 6 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{24 b x^{7}} + \frac{c^{2} \left (A c - 6 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{16 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**10,x)

[Out]

-A*(b*x**2 + c*x**4)**(5/2)/(6*b*x**11) + c*(A*c - 6*B*b)*sqrt(b*x**2 + c*x**4)/
(16*b*x**3) + (A*c - 6*B*b)*(b*x**2 + c*x**4)**(3/2)/(24*b*x**7) + c**2*(A*c - 6
*B*b)*atanh(sqrt(b)*x/sqrt(b*x**2 + c*x**4))/(16*b**(3/2))

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Mathematica [A]  time = 0.315237, size = 151, normalized size = 1.08 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{b} \sqrt{b+c x^2} \left (A \left (8 b^2+14 b c x^2+3 c^2 x^4\right )+6 b B x^2 \left (2 b+5 c x^2\right )\right )+3 c^2 x^6 \log (x) (A c-6 b B)-3 c^2 x^6 (A c-6 b B) \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )\right )}{48 b^{3/2} x^7 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^10,x]

[Out]

-(Sqrt[x^2*(b + c*x^2)]*(Sqrt[b]*Sqrt[b + c*x^2]*(6*b*B*x^2*(2*b + 5*c*x^2) + A*
(8*b^2 + 14*b*c*x^2 + 3*c^2*x^4)) + 3*c^2*(-6*b*B + A*c)*x^6*Log[x] - 3*c^2*(-6*
b*B + A*c)*x^6*Log[b + Sqrt[b]*Sqrt[b + c*x^2]]))/(48*b^(3/2)*x^7*Sqrt[b + c*x^2
])

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Maple [B]  time = 0.02, size = 273, normalized size = 2. \[ -{\frac{1}{48\,{x}^{9}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -18\,B{c}^{2}\sqrt{c{x}^{2}+b}{x}^{6}{b}^{7/2}-6\,B{c}^{2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{6}{b}^{5/2}+3\,A{c}^{3}\sqrt{c{x}^{2}+b}{x}^{6}{b}^{5/2}+A{c}^{3} \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{6}{b}^{{\frac{3}{2}}}+6\,Bc \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{5/2}-A{c}^{2} \left ( c{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{4}{b}^{{\frac{3}{2}}}+12\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{7/2}-2\,Ac \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{5/2}-3\,A{c}^{3}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{6}{b}^{3}+18\,B{c}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{6}{b}^{4}+8\,A \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{7/2} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^10,x)

[Out]

-1/48*(c*x^4+b*x^2)^(3/2)*(-18*B*c^2*(c*x^2+b)^(1/2)*x^6*b^(7/2)-6*B*c^2*(c*x^2+
b)^(3/2)*x^6*b^(5/2)+3*A*c^3*(c*x^2+b)^(1/2)*x^6*b^(5/2)+A*c^3*(c*x^2+b)^(3/2)*x
^6*b^(3/2)+6*B*c*(c*x^2+b)^(5/2)*x^4*b^(5/2)-A*c^2*(c*x^2+b)^(5/2)*x^4*b^(3/2)+1
2*B*(c*x^2+b)^(5/2)*x^2*b^(7/2)-2*A*c*(c*x^2+b)^(5/2)*x^2*b^(5/2)-3*A*c^3*ln(2*(
b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*x^6*b^3+18*B*c^2*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/
x)*x^6*b^4+8*A*(c*x^2+b)^(5/2)*b^(7/2))/x^9/(c*x^2+b)^(3/2)/b^(9/2)

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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((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274411, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B b c^{2} - A c^{3}\right )} \sqrt{b} x^{7} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \,{\left (3 \,{\left (10 \, B b^{2} c + A b c^{2}\right )} x^{4} + 8 \, A b^{3} + 2 \,{\left (6 \, B b^{3} + 7 \, A b^{2} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, b^{2} x^{7}}, \frac{3 \,{\left (6 \, B b c^{2} - A c^{3}\right )} \sqrt{-b} x^{7} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) -{\left (3 \,{\left (10 \, B b^{2} c + A b c^{2}\right )} x^{4} + 8 \, A b^{3} + 2 \,{\left (6 \, B b^{3} + 7 \, A b^{2} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, b^{2} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^10,x, algorithm="fricas")

[Out]

[-1/96*(3*(6*B*b*c^2 - A*c^3)*sqrt(b)*x^7*log(-((c*x^3 + 2*b*x)*sqrt(b) + 2*sqrt
(c*x^4 + b*x^2)*b)/x^3) + 2*(3*(10*B*b^2*c + A*b*c^2)*x^4 + 8*A*b^3 + 2*(6*B*b^3
+ 7*A*b^2*c)*x^2)*sqrt(c*x^4 + b*x^2))/(b^2*x^7), 1/48*(3*(6*B*b*c^2 - A*c^3)*s
qrt(-b)*x^7*arctan(sqrt(-b)*x/sqrt(c*x^4 + b*x^2)) - (3*(10*B*b^2*c + A*b*c^2)*x
^4 + 8*A*b^3 + 2*(6*B*b^3 + 7*A*b^2*c)*x^2)*sqrt(c*x^4 + b*x^2))/(b^2*x^7)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**10,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**10, x)

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GIAC/XCAS [A]  time = 0.292879, size = 236, normalized size = 1.69 \[ \frac{\frac{3 \,{\left (6 \, B b c^{3}{\rm sign}\left (x\right ) - A c^{4}{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{30 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} B b c^{3}{\rm sign}\left (x\right ) - 48 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B b^{2} c^{3}{\rm sign}\left (x\right ) + 18 \, \sqrt{c x^{2} + b} B b^{3} c^{3}{\rm sign}\left (x\right ) + 3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} A c^{4}{\rm sign}\left (x\right ) + 8 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A b c^{4}{\rm sign}\left (x\right ) - 3 \, \sqrt{c x^{2} + b} A b^{2} c^{4}{\rm sign}\left (x\right )}{b c^{3} x^{6}}}{48 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^10,x, algorithm="giac")

[Out]

1/48*(3*(6*B*b*c^3*sign(x) - A*c^4*sign(x))*arctan(sqrt(c*x^2 + b)/sqrt(-b))/(sq
rt(-b)*b) - (30*(c*x^2 + b)^(5/2)*B*b*c^3*sign(x) - 48*(c*x^2 + b)^(3/2)*B*b^2*c
^3*sign(x) + 18*sqrt(c*x^2 + b)*B*b^3*c^3*sign(x) + 3*(c*x^2 + b)^(5/2)*A*c^4*si
gn(x) + 8*(c*x^2 + b)^(3/2)*A*b*c^4*sign(x) - 3*sqrt(c*x^2 + b)*A*b^2*c^4*sign(x
))/(b*c^3*x^6))/c