3.729 \(\int \frac{1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=211 \[ -\frac{b^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 (b c-a d)^{5/2}}-\frac{d \left (5 a^2 d^2-8 a b c d+b^2 c^2\right )}{2 a c^3 \sqrt{c+d x^2} (b c-a d)^2}+\frac{(5 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{7/2}}-\frac{d (3 b c-5 a d)}{6 a c^2 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left (c+d x^2\right )^{3/2}} \]

[Out]

-(d*(3*b*c - 5*a*d))/(6*a*c^2*(b*c - a*d)*(c + d*x^2)^(3/2)) - 1/(2*a*c*x^2*(c +
d*x^2)^(3/2)) - (d*(b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2))/(2*a*c^3*(b*c - a*d)^2*Sq
rt[c + d*x^2]) + ((2*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(7/
2)) - (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a^2*(b*c - a
*d)^(5/2))

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Rubi [A]  time = 0.948776, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{b^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 (b c-a d)^{5/2}}-\frac{d \left (5 a^2 d^2-8 a b c d+b^2 c^2\right )}{2 a c^3 \sqrt{c+d x^2} (b c-a d)^2}+\frac{(5 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{7/2}}-\frac{d (3 b c-5 a d)}{6 a c^2 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^2)*(c + d*x^2)^(5/2)),x]

[Out]

-(d*(3*b*c - 5*a*d))/(6*a*c^2*(b*c - a*d)*(c + d*x^2)^(3/2)) - 1/(2*a*c*x^2*(c +
d*x^2)^(3/2)) - (d*(b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2))/(2*a*c^3*(b*c - a*d)^2*Sq
rt[c + d*x^2]) + ((2*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(7/
2)) - (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a^2*(b*c - a
*d)^(5/2))

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Rubi in Sympy [A]  time = 119.089, size = 189, normalized size = 0.9 \[ - \frac{1}{2 a c x^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} - \frac{d \left (5 a d - 3 b c\right )}{6 a c^{2} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{d \left (5 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right )}{2 a c^{3} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{\left (5 a d + 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{2} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**2+a)/(d*x**2+c)**(5/2),x)

[Out]

-1/(2*a*c*x**2*(c + d*x**2)**(3/2)) - d*(5*a*d - 3*b*c)/(6*a*c**2*(c + d*x**2)**
(3/2)*(a*d - b*c)) - d*(5*a**2*d**2 - 8*a*b*c*d + b**2*c**2)/(2*a*c**3*sqrt(c +
d*x**2)*(a*d - b*c)**2) + b**(7/2)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c)
)/(a**2*(a*d - b*c)**(5/2)) + (5*a*d + 2*b*c)*atanh(sqrt(c + d*x**2)/sqrt(c))/(2
*a**2*c**(7/2))

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Mathematica [C]  time = 3.13942, size = 409, normalized size = 1.94 \[ \frac{1}{2} \left (-\frac{b^{7/2} \log \left (\frac{2 a^2 (b c-a d) \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^4 x+i \sqrt{a} b^{7/2}}\right )}{a^2 (b c-a d)^{5/2}}-\frac{b^{7/2} \log \left (\frac{2 a^2 (b c-a d) \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^4 x-i \sqrt{a} b^{7/2}}\right )}{a^2 (b c-a d)^{5/2}}+\frac{(5 a d+2 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^2 c^{7/2}}-\frac{\log (x) (5 a d+2 b c)}{a^2 c^{7/2}}+\frac{\sqrt{c+d x^2} \left (\frac{6 d^2 (3 b c-2 a d)}{\left (c+d x^2\right ) (b c-a d)^2}+\frac{2 c d^2}{\left (c+d x^2\right )^2 (b c-a d)}-\frac{3}{a x^2}\right )}{3 c^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^2)*(c + d*x^2)^(5/2)),x]

[Out]

((Sqrt[c + d*x^2]*(-3/(a*x^2) + (2*c*d^2)/((b*c - a*d)*(c + d*x^2)^2) + (6*d^2*(
3*b*c - 2*a*d))/((b*c - a*d)^2*(c + d*x^2))))/(3*c^3) - ((2*b*c + 5*a*d)*Log[x])
/(a^2*c^(7/2)) + ((2*b*c + 5*a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/(a^2*c^(7/2)
) - (b^(7/2)*Log[(2*a^2*(b*c - a*d)*(Sqrt[b]*c*Sqrt[b*c - a*d] - I*Sqrt[a]*d*Sqr
t[b*c - a*d]*x + b*c*Sqrt[c + d*x^2] - a*d*Sqrt[c + d*x^2]))/(I*Sqrt[a]*b^(7/2)
+ b^4*x)])/(a^2*(b*c - a*d)^(5/2)) - (b^(7/2)*Log[(2*a^2*(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^(7/2) + b^4*x)])/(a^2*(b*c - a*d)^(5/2)))/2

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Maple [B]  time = 0.022, size = 1289, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2+a)/(d*x^2+c)^(5/2),x)

[Out]

-1/2/a/c/x^2/(d*x^2+c)^(3/2)-5/6/a*d/c^2/(d*x^2+c)^(3/2)-5/2/a*d/c^3/(d*x^2+c)^(
1/2)+5/2/a*d/c^(7/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-1/3*b/a^2/c/(d*x^2+c)
^(3/2)-b/a^2/c^2/(d*x^2+c)^(1/2)+b/a^2/c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2)
)/x)-1/6*b^2/a^2/(a*d-b*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)+1/6*b/a^2*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*
b/a^2*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*b^3/a^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*b^2/a
^2/(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*b^3/a^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*b^2/a^2/(a*d-b*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)-1/6*b/a^2*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*b/a^2*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*b^3/a
^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*b^2/a^2/(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*b^3/a^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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^3), x)

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Fricas [A]  time = 2.31912, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[1/12*(3*(b^3*c^3*d^2*x^6 + 2*b^3*c^4*d*x^4 + b^3*c^5*x^2)*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)) - 2*(3*a*b^2*c^4 - 6
*a^2*b*c^3*d + 3*a^3*c^2*d^2 + 3*(a*b^2*c^2*d^2 - 8*a^2*b*c*d^3 + 5*a^3*d^4)*x^4
+ 2*(3*a*b^2*c^3*d - 16*a^2*b*c^2*d^2 + 10*a^3*c*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt
(c) + 3*((2*b^3*c^3*d^2 + a*b^2*c^2*d^3 - 8*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 2*(2*
b^3*c^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^4 + (2*b^3*c^5 + a*
b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) +
2*sqrt(d*x^2 + c)*c)/x^2))/(((a^2*b^2*c^5*d^2 - 2*a^3*b*c^4*d^3 + a^4*c^3*d^4)*x
^6 + 2*(a^2*b^2*c^6*d - 2*a^3*b*c^5*d^2 + a^4*c^4*d^3)*x^4 + (a^2*b^2*c^7 - 2*a^
3*b*c^6*d + a^4*c^5*d^2)*x^2)*sqrt(c)), 1/12*(3*(b^3*c^3*d^2*x^6 + 2*b^3*c^4*d*x
^4 + b^3*c^5*x^2)*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)) - 2*(3*a*b^2*c^4 - 6*a^2*b*c^3*d + 3*a^3*c^2*d^2 + 3*(a*b^2*
c^2*d^2 - 8*a^2*b*c*d^3 + 5*a^3*d^4)*x^4 + 2*(3*a*b^2*c^3*d - 16*a^2*b*c^2*d^2 +
10*a^3*c*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) + 6*((2*b^3*c^3*d^2 + a*b^2*c^2*d^3
- 8*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 2*(2*b^3*c^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2
*d^3 + 5*a^3*c*d^4)*x^4 + (2*b^3*c^5 + a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3*c^2
*d^3)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/(((a^2*b^2*c^5*d^2 - 2*a^3*b*c^4*d^
3 + a^4*c^3*d^4)*x^6 + 2*(a^2*b^2*c^6*d - 2*a^3*b*c^5*d^2 + a^4*c^4*d^3)*x^4 + (
a^2*b^2*c^7 - 2*a^3*b*c^6*d + a^4*c^5*d^2)*x^2)*sqrt(-c)), 1/12*(6*(b^3*c^3*d^2*
x^6 + 2*b^3*c^4*d*x^4 + b^3*c^5*x^2)*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*(3
*a*b^2*c^4 - 6*a^2*b*c^3*d + 3*a^3*c^2*d^2 + 3*(a*b^2*c^2*d^2 - 8*a^2*b*c*d^3 +
5*a^3*d^4)*x^4 + 2*(3*a*b^2*c^3*d - 16*a^2*b*c^2*d^2 + 10*a^3*c*d^3)*x^2)*sqrt(d
*x^2 + c)*sqrt(c) + 3*((2*b^3*c^3*d^2 + a*b^2*c^2*d^3 - 8*a^2*b*c*d^4 + 5*a^3*d^
5)*x^6 + 2*(2*b^3*c^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^4 + (
2*b^3*c^5 + a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x^2)*log(-((d*x^2 + 2
*c)*sqrt(c) + 2*sqrt(d*x^2 + c)*c)/x^2))/(((a^2*b^2*c^5*d^2 - 2*a^3*b*c^4*d^3 +
a^4*c^3*d^4)*x^6 + 2*(a^2*b^2*c^6*d - 2*a^3*b*c^5*d^2 + a^4*c^4*d^3)*x^4 + (a^2*
b^2*c^7 - 2*a^3*b*c^6*d + a^4*c^5*d^2)*x^2)*sqrt(c)), 1/6*(3*(b^3*c^3*d^2*x^6 +
2*b^3*c^4*d*x^4 + b^3*c^5*x^2)*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)))) - (3*a*b^2*
c^4 - 6*a^2*b*c^3*d + 3*a^3*c^2*d^2 + 3*(a*b^2*c^2*d^2 - 8*a^2*b*c*d^3 + 5*a^3*d
^4)*x^4 + 2*(3*a*b^2*c^3*d - 16*a^2*b*c^2*d^2 + 10*a^3*c*d^3)*x^2)*sqrt(d*x^2 +
c)*sqrt(-c) + 3*((2*b^3*c^3*d^2 + a*b^2*c^2*d^3 - 8*a^2*b*c*d^4 + 5*a^3*d^5)*x^6
+ 2*(2*b^3*c^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^4 + (2*b^3*
c^5 + a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x^2)*arctan(sqrt(-c)/sqrt(d
*x^2 + c)))/(((a^2*b^2*c^5*d^2 - 2*a^3*b*c^4*d^3 + a^4*c^3*d^4)*x^6 + 2*(a^2*b^2
*c^6*d - 2*a^3*b*c^5*d^2 + a^4*c^4*d^3)*x^4 + (a^2*b^2*c^7 - 2*a^3*b*c^6*d + a^4
*c^5*d^2)*x^2)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**2+a)/(d*x**2+c)**(5/2),x)

[Out]

Integral(1/(x**3*(a + b*x**2)*(c + d*x**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.248098, size = 294, normalized size = 1.39 \[ \frac{1}{6} \,{\left (\frac{6 \, b^{4} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (9 \,{\left (d x^{2} + c\right )} b c + b c^{2} - 6 \,{\left (d x^{2} + c\right )} a d - a c d\right )}}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{3 \,{\left (2 \, b c + 5 \, a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{3} d^{2}} - \frac{3 \, \sqrt{d x^{2} + c}}{a c^{3} d^{2} x^{2}}\right )} d^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^3),x, algorithm="giac")

[Out]

1/6*(6*b^4*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a^2*b^2*c^2*d^2 - 2*
a^3*b*c*d^3 + a^4*d^4)*sqrt(-b^2*c + a*b*d)) + 2*(9*(d*x^2 + c)*b*c + b*c^2 - 6*
(d*x^2 + c)*a*d - a*c*d)/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*(d*x^2 + c)^(3/2
)) - 3*(2*b*c + 5*a*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^2*sqrt(-c)*c^3*d^2) -
3*sqrt(d*x^2 + c)/(a*c^3*d^2*x^2))*d^2