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

Optimal. Leaf size=144 \[ -\frac{(b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2}+\frac{d \sqrt{c+d x^2} (2 a d+b c)}{2 a b}-\frac{c \left (c+d x^2\right )^{3/2}}{2 a x^2} \]

[Out]

(d*(b*c + 2*a*d)*Sqrt[c + d*x^2])/(2*a*b) - (c*(c + d*x^2)^(3/2))/(2*a*x^2) + (c
^(3/2)*(2*b*c - 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2) - ((b*c - a*d)^
(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a^2*b^(3/2))

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Rubi [A]  time = 0.638936, antiderivative size = 144, 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 \[ -\frac{(b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2}+\frac{d \sqrt{c+d x^2} (2 a d+b c)}{2 a b}-\frac{c \left (c+d x^2\right )^{3/2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*c + 2*a*d)*Sqrt[c + d*x^2])/(2*a*b) - (c*(c + d*x^2)^(3/2))/(2*a*x^2) + (c
^(3/2)*(2*b*c - 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2) - ((b*c - a*d)^
(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a^2*b^(3/2))

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Rubi in Sympy [A]  time = 69.7412, size = 126, normalized size = 0.88 \[ - \frac{c \left (c + d x^{2}\right )^{\frac{3}{2}}}{2 a x^{2}} + \frac{d \sqrt{c + d x^{2}} \left (2 a d + b c\right )}{2 a b} - \frac{c^{\frac{3}{2}} \left (5 a d - 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{2}} - \frac{\left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.654135, size = 311, normalized size = 2.16 \[ \frac{1}{2} \left (-\frac{(b c-a d)^{5/2} \log \left (\frac{2 a^2 b^{3/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{7/2}}\right )}{a^2 b^{3/2}}-\frac{(b c-a d)^{5/2} \log \left (\frac{2 a^2 b^{3/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{7/2}}\right )}{a^2 b^{3/2}}+\frac{c^{3/2} (2 b c-5 a d) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^2}+\frac{c^{3/2} \log (x) (5 a d-2 b c)}{a^2}+2 \sqrt{c+d x^2} \left (\frac{d^2}{b}-\frac{c^2}{2 a x^2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.022, size = 3247, normalized size = 22.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/b^2*a/(-(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)))*d^3-3/2/b/(-(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)))*d^2*c+3/2/a/(-(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)))*d*c^2-1/2*b/a^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)))*c^3-1/8/a^2*d*(-a*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)^(3/2)*x-15/16/a^2*d^(1/2)*(-a*b)^(1/2)*ln(
(-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(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))*c^2+1/2/b^2*a/(-(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)))*d^3+1/8/a^2*d*(-a*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)^(3/2)*x+1/10
*b/a^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)^(5/2)-1/6/a*((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)*d-7/16/a^2*d*(-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+1/4/b/a*d^2*(-a*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-5/4/b/a*d^(3/2)*(-a*b)^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))
*d)/d^(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))*c+5/4/b/a*d^(3/2)*(-a*b)^(1/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-
a*b)^(1/2))*d)/d^(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))*c+7/16/a^2*d*(-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-1/4/b/a*d^2*(-a*
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-3/2/b/(-(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)))*d^2*c+3
/2/a/(-(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)))*d*c^2-1/2*b/a^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)))*c^3-1/3*b/a^2*c*(d*x^2+c)^(3/2)+b/a^2
*c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-b/a^2*(d*x^2+c)^(1/2)*c^2+1/6*b/a
^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
)^(3/2)*c-1/2/a/c/x^2*(d*x^2+c)^(7/2)+1/2/a*d/c*(d*x^2+c)^(5/2)-5/2/a*d*c^(3/2)*
ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+5/2/a*d*c*(d*x^2+c)^(1/2)+1/6*b/a^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)^(3/2)*c
-1/a*((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)*d*c-1/a*((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)*d*c+1/2*b/a^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)*c^2-1/2/b^2*d^(5/2)*(-a*b)^(1/2)*ln((-
d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(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))+1/2*b/a^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)*c^2+1/2/b^
2*d^(5/2)*(-a*b)^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(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)
)+1/2/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)^(1/2)*d^2+1/10*b/a^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)^(5/2)-1/6/a*((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)*d+1/2/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)^(1/2)*d^2+5/6/a*d*(d*x^2+c)^(
3/2)-1/5*b/a^2*(d*x^2+c)^(5/2)+15/16/a^2*d^(1/2)*(-a*b)^(1/2)*ln((d*(-a*b)^(1/2)
/b+(x-1/b*(-a*b)^(1/2))*d)/d^(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))*c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.50896, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt((b*c - a*d)/b)*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*(b^2*d*x^2
+ 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a
^2)) - (2*b^2*c^2 - 5*a*b*c*d)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(
c) + 2*c)/x^2) + 2*(2*a^2*d^2*x^2 - a*b*c^2)*sqrt(d*x^2 + c))/(a^2*b*x^2), 1/4*(
2*(2*b^2*c^2 - 5*a*b*c*d)*sqrt(-c)*x^2*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) + (b
^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt((b*c - a*d)/b)*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*(b^2*d*x^2 + 2*b^2*
c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2
*(2*a^2*d^2*x^2 - a*b*c^2)*sqrt(d*x^2 + c))/(a^2*b*x^2), -1/4*(2*(b^2*c^2 - 2*a*
b*c*d + a^2*d^2)*x^2*sqrt(-(b*c - a*d)/b)*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)/(sq
rt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b))) + (2*b^2*c^2 - 5*a*b*c*d)*sqrt(c)*x^2*log
(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(2*a^2*d^2*x^2 - a*b*c^2)*s
qrt(d*x^2 + c))/(a^2*b*x^2), 1/2*((2*b^2*c^2 - 5*a*b*c*d)*sqrt(-c)*x^2*arctan(c/
(sqrt(d*x^2 + c)*sqrt(-c))) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sqrt(-(b*c - a
*d)/b)*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)/(sqrt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b
))) + (2*a^2*d^2*x^2 - a*b*c^2)*sqrt(d*x^2 + c))/(a^2*b*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241429, size = 231, normalized size = 1.6 \[ \frac{1}{2} \, d^{2}{\left (\frac{2 \, \sqrt{d x^{2} + c}}{b} - \frac{\sqrt{d x^{2} + c} c^{2}}{a d^{2} x^{2}} - \frac{{\left (2 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} + \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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