3.727 \(\int \frac{1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=145 \[ \frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a (b c-a d)^{5/2}}-\frac{d (2 b c-a d)}{c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a c^{5/2}} \]
[Out]
-d/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d))/(c^2*(b*c - a*d)^2*Sq
rt[c + d*x^2]) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a*c^(5/2)) + (b^(5/2)*ArcTanh
[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*(b*c - a*d)^(5/2))
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Rubi [A] time = 0.588484, antiderivative size = 145, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292
\[ \frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a (b c-a d)^{5/2}}-\frac{d (2 b c-a d)}{c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a c^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x^2)*(c + d*x^2)^(5/2)),x]
[Out]
-d/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d))/(c^2*(b*c - a*d)^2*Sq
rt[c + d*x^2]) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a*c^(5/2)) + (b^(5/2)*ArcTanh
[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*(b*c - a*d)^(5/2))
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Rubi in Sympy [A] time = 76.2553, size = 121, normalized size = 0.83 \[ \frac{d}{3 c \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{d \left (a d - 2 b c\right )}{c^{2} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a \left (a d - b c\right )^{\frac{5}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x**2+a)/(d*x**2+c)**(5/2),x)
[Out]
d/(3*c*(c + d*x**2)**(3/2)*(a*d - b*c)) + d*(a*d - 2*b*c)/(c**2*sqrt(c + d*x**2)
*(a*d - b*c)**2) - b**(5/2)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/(a*(a
*d - b*c)**(5/2)) - atanh(sqrt(c + d*x**2)/sqrt(c))/(a*c**(5/2))
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Mathematica [C] time = 1.81088, size = 365, normalized size = 2.52 \[ \frac{1}{6} \left (\frac{3 b^{5/2} \log \left (-\frac{2 a (b c-a d) \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^3 x+i \sqrt{a} b^{5/2}}\right )}{a (b c-a d)^{5/2}}+\frac{3 b^{5/2} \log \left (-\frac{2 a (b c-a d) \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^3 x-i \sqrt{a} b^{5/2}}\right )}{a (b c-a d)^{5/2}}+\frac{6 d (a d-2 b c)}{c^2 \sqrt{c+d x^2} (b c-a d)^2}+\frac{2 d}{c \left (c+d x^2\right )^{3/2} (a d-b c)}-\frac{6 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a c^{5/2}}+\frac{6 \log (x)}{a c^{5/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*x^2)*(c + d*x^2)^(5/2)),x]
[Out]
((2*d)/(c*(-(b*c) + a*d)*(c + d*x^2)^(3/2)) + (6*d*(-2*b*c + a*d))/(c^2*(b*c - a
*d)^2*Sqrt[c + d*x^2]) + (6*Log[x])/(a*c^(5/2)) - (6*Log[c + Sqrt[c]*Sqrt[c + d*
x^2]])/(a*c^(5/2)) + (3*b^(5/2)*Log[(-2*a*(b*c - a*d)*(Sqrt[b]*c*Sqrt[b*c - a*d]
- I*Sqrt[a]*d*Sqrt[b*c - a*d]*x + b*c*Sqrt[c + d*x^2] - a*d*Sqrt[c + d*x^2]))/(
I*Sqrt[a]*b^(5/2) + b^3*x)])/(a*(b*c - a*d)^(5/2)) + (3*b^(5/2)*Log[(-2*a*(b*c -
a*d)*(Sqrt[b]*c*Sqrt[b*c - a*d] + I*Sqrt[a]*d*Sqrt[b*c - a*d]*x + b*c*Sqrt[c +
d*x^2] - a*d*Sqrt[c + d*x^2]))/((-I)*Sqrt[a]*b^(5/2) + b^3*x)])/(a*(b*c - a*d)^(
5/2)))/6
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Maple [B] time = 0.019, size = 1186, normalized size = 8.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x^2+a)/(d*x^2+c)^(5/2),x)
[Out]
1/3/a/c/(d*x^2+c)^(3/2)+1/a/c^2/(d*x^2+c)^(1/2)-1/a/c^(5/2)*ln((2*c+2*c^(1/2)*(d
*x^2+c)^(1/2))/x)+1/6/a/(a*d-b*c)*b/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b
*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-1/6/a*d*(-a*b)^(1/2)/(a*d-b*c)/c/((x-1/
b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x
-1/3/a*d*(-a*b)^(1/2)/(a*d-b*c)/c^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b
*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-1/2/a*b^2/(a*d-b*c)^2/((x-1/b*(-a*b)^
(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/2/a*b/(a
*d-b*c)^2*(-a*b)^(1/2)/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a
*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d+1/2/a*b^2/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln(
(-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*(
(x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1
/2))/(x-1/b*(-a*b)^(1/2)))+1/6/a/(a*d-b*c)*b/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b
)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+1/6/a*d*(-a*b)^(1/2)/(a*d-b*c)
/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(3/2)*x+1/3/a*d*(-a*b)^(1/2)/(a*d-b*c)/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b
)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-1/2/a*b^2/(a*d-b*c)^2/((x+1/
b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1
/2/a*b/(a*d-b*c)^2*(-a*b)^(1/2)/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(
x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d+1/2/a*b^2/(a*d-b*c)^2/(-(a*d-b*c)/b)^
(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b
)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b
*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x),x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x), x)
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Fricas [A] time = 1.21486, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x),x, algorithm="fricas")
[Out]
[1/12*(3*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(c)*sqrt(b/(b*c - a*d
))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2
)*x^2 + 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2
+ c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(7*a*b*c^2*d - 4*a^2
*c*d^2 + 3*(2*a*b*c*d^2 - a^2*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(c) + 6*(b^2*c^4 - 2
*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^4 + 2*(b^2*c^
3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2
+ c)*c)/x^2))/((a*b^2*c^6 - 2*a^2*b*c^5*d + a^3*c^4*d^2 + (a*b^2*c^4*d^2 - 2*a^
2*b*c^3*d^3 + a^3*c^2*d^4)*x^4 + 2*(a*b^2*c^5*d - 2*a^2*b*c^4*d^2 + a^3*c^3*d^3)
*x^2)*sqrt(c)), 1/12*(3*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(-c)*s
qrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2
*c*d - 3*a*b*d^2)*x^2 + 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)
*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(7*a
*b*c^2*d - 4*a^2*c*d^2 + 3*(2*a*b*c*d^2 - a^2*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(-c)
- 12*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^
4)*x^4 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2)*arctan(sqrt(-c)/sqrt(d*x
^2 + c)))/((a*b^2*c^6 - 2*a^2*b*c^5*d + a^3*c^4*d^2 + (a*b^2*c^4*d^2 - 2*a^2*b*c
^3*d^3 + a^3*c^2*d^4)*x^4 + 2*(a*b^2*c^5*d - 2*a^2*b*c^4*d^2 + a^3*c^3*d^3)*x^2)
*sqrt(-c)), -1/6*(3*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(c)*sqrt(-
b/(b*c - a*d))*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)/(sqrt(d*x^2 + c)*(b*c - a*d)*
sqrt(-b/(b*c - a*d)))) + 2*(7*a*b*c^2*d - 4*a^2*c*d^2 + 3*(2*a*b*c*d^2 - a^2*d^3
)*x^2)*sqrt(d*x^2 + c)*sqrt(c) - 3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c
^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^4 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*
x^2)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/((a*b^2*c^6 - 2*a^
2*b*c^5*d + a^3*c^4*d^2 + (a*b^2*c^4*d^2 - 2*a^2*b*c^3*d^3 + a^3*c^2*d^4)*x^4 +
2*(a*b^2*c^5*d - 2*a^2*b*c^4*d^2 + a^3*c^3*d^3)*x^2)*sqrt(c)), -1/6*(3*(b^2*c^2*
d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(-c)*sqrt(-b/(b*c - a*d))*arctan(-1/2*(
b*d*x^2 + 2*b*c - a*d)/(sqrt(d*x^2 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d)))) + 2*(
7*a*b*c^2*d - 4*a^2*c*d^2 + 3*(2*a*b*c*d^2 - a^2*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(
-c) + 6*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*
d^4)*x^4 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x^2)*arctan(sqrt(-c)/sqrt(d
*x^2 + c)))/((a*b^2*c^6 - 2*a^2*b*c^5*d + a^3*c^4*d^2 + (a*b^2*c^4*d^2 - 2*a^2*b
*c^3*d^3 + a^3*c^2*d^4)*x^4 + 2*(a*b^2*c^5*d - 2*a^2*b*c^4*d^2 + a^3*c^3*d^3)*x^
2)*sqrt(-c))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x**2+a)/(d*x**2+c)**(5/2),x)
[Out]
Integral(1/(x*(a + b*x**2)*(c + d*x**2)**(5/2)), x)
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GIAC/XCAS [A] time = 0.236478, size = 242, normalized size = 1.67 \[ -\frac{1}{3} \,{\left (\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} + \frac{6 \,{\left (d x^{2} + c\right )} b c + b c^{2} - 3 \,{\left (d x^{2} + c\right )} a d - a c d}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{3 \, \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} c^{2} d}\right )} d \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x),x, algorithm="giac")
[Out]
-1/3*(3*b^3*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a*b^2*c^2*d - 2*a^2
*b*c*d^2 + a^3*d^3)*sqrt(-b^2*c + a*b*d)) + (6*(d*x^2 + c)*b*c + b*c^2 - 3*(d*x^
2 + c)*a*d - a*c*d)/((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*(d*x^2 + c)^(3/2)) -
3*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a*sqrt(-c)*c^2*d))*d