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

Optimal. Leaf size=126 \[ -\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{7/2}}+\frac{5 d \sqrt{c+d x^2} (b c-a d)}{2 b^3}-\frac{\left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 d \left (c+d x^2\right )^{3/2}}{6 b^2} \]

[Out]

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

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Rubi [A]  time = 0.266577, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{7/2}}+\frac{5 d \sqrt{c+d x^2} (b c-a d)}{2 b^3}-\frac{\left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 d \left (c+d x^2\right )^{3/2}}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.1669, size = 110, normalized size = 0.87 \[ - \frac{\left (c + d x^{2}\right )^{\frac{5}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{5 d \left (c + d x^{2}\right )^{\frac{3}{2}}}{6 b^{2}} - \frac{5 d \sqrt{c + d x^{2}} \left (a d - b c\right )}{2 b^{3}} + \frac{5 d \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x**2)**(5/2)/(2*b*(a + b*x**2)) + 5*d*(c + d*x**2)**(3/2)/(6*b**2) - 5*d
*sqrt(c + d*x**2)*(a*d - b*c)/(2*b**3) + 5*d*(a*d - b*c)**(3/2)*atan(sqrt(b)*sqr
t(c + d*x**2)/sqrt(a*d - b*c))/(2*b**(7/2))

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Mathematica [C]  time = 0.674033, size = 289, normalized size = 2.29 \[ \frac{-15 d (b c-a d)^{3/2} \log \left (\frac{4 b^{7/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{5 d \left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{5/2}}\right )-15 d (b c-a d)^{3/2} \log \left (\frac{4 b^{7/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{5 d \left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{5/2}}\right )-\frac{2 \sqrt{b} \sqrt{c+d x^2} \left (2 d \left (a+b x^2\right ) (6 a d-7 b c)+3 (b c-a d)^2-2 b d^2 x^2 \left (a+b x^2\right )\right )}{a+b x^2}}{12 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[b]*Sqrt[c + d*x^2]*(3*(b*c - a*d)^2 + 2*d*(-7*b*c + 6*a*d)*(a + b*x^2)
- 2*b*d^2*x^2*(a + b*x^2)))/(a + b*x^2) - 15*d*(b*c - a*d)^(3/2)*Log[(4*b^(7/2)
*(Sqrt[b]*c - I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(5*d*(b*c - a*d)
^(5/2)*(I*Sqrt[a] + Sqrt[b]*x))] - 15*d*(b*c - a*d)^(3/2)*Log[(4*b^(7/2)*(Sqrt[b
]*c + I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(5*d*(b*c - a*d)^(5/2)*(
(-I)*Sqrt[a] + Sqrt[b]*x))])/(12*b^(7/2))

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Maple [B]  time = 0.023, size = 4363, normalized size = 34.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/12*a/b^2*d^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)-5/12/b*d/(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)*c+5/12*a/b^2*d^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)-5/12/b*d/(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)*c-5/4*a^2/b^3*d^3/(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)^(1/2)-5/4/b*d/(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)^
(1/2)*c^2-5/4/b*d/(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)^(1/2)*c^2-5/4*a^2/b^3*d^3/(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)^(1/2)+5/16*(-a*b)^
(1/2)/a/b*d/(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-5/16*(-a*b)^(1/2)/b^2*d^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)*x-75/32*
(-a*b)^(1/2)/b^2*d^(3/2)/(a*d-b*c)*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+5/2*a/b^2*d^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)^(1/2)*c-5/4*(-a*b)^(1/2)*a^2/b^4*d^(7/2
)/(a*d-b*c)*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))-5/4*a^3/b
^4*d^4/(a*d-b*c)/(-(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)))+5/4/b*d/(a*d
-b*c)/(-(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+1/4*(-a*b)^(1/2)/a/
b/(a*d-b*c)/(x+1/b*(-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)^(7/2)+5/16*(-a*b)^(1/2)/b^2*d^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)*x
+75/32*(-a*b)^(1/2)/b^2*d^(3/2)/(a*d-b*c)*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+5/2*a/b^2*d^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)^(1/2)*c+5/4*(-a*b)^(1/2)*a^2/b^
4*d^(7/2)/(a*d-b*c)*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))-
5/4*a^3/b^4*d^4/(a*d-b*c)/(-(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)))+5/4
/b*d/(a*d-b*c)/(-(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^3-1/4*(-a*b)
^(1/2)/a/b/(a*d-b*c)/(x-1/b*(-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)^(7/2)-1/4/b*d/(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)^(5/2)-1/4/b*d/(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)^(5/2)+35/32*(-a*b)^(1/2)/b^2*d^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)^(1/2)*x-5/8*(-a*b)^(1/2)*a/b^
3*d^3/(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)^(1/2)*x-25/8*(-a*b)^(1/2)*a/b^3*d^(5/2)/(a*d-b*c)*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+15/4*a^2/b^3*d^3/(a*d-b*c)/(-(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-15/4*a/b^2*d^2/(a*d-b*c)/(-(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^2+1/4*(-a*b)^(1/2)/a/b*d/(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)^(
5/2)*x+15/32*(-a*b)^(1/2)/a/b*d^(1/2)/(a*d-b*c)*c^3*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))-35/32*(-a*b)^(1/2)/b^2*d^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)^(1/2)*x+5/8
*(-a*b)^(1/2)*a/b^3*d^3/(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)^(1/2)*x+25/8*(-a*b)^(1/2)*a/b^3*d^(5/2)/(a*d-b*
c)*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+15/4*a^2/b^3*d^3
/(a*d-b*c)/(-(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-15/4*a/b^2*d^2/(
a*d-b*c)/(-(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^2-1/4*(-a*b)^(1/2)
/a/b*d/(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)^(5/2)*x-15/32*(-a*b)^(1/2)/a/b*d^(1/2)/(a*d-b*c)*c^3*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))+15/32*(-a*b)^(1/2)/a/b*d/(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-5/16*(-a*b)^(1/2)/a/b*d/(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-15/32*(-a*b)^(1/2)/a/b
*d/(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

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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((d*x^2 + c)^(5/2)*x/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286367, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \,{\left (2 \, b^{2} d^{2} x^{4} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{24 \,{\left (b^{4} x^{2} + a b^{3}\right )}}, -\frac{15 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) - 2 \,{\left (2 \, b^{2} d^{2} x^{4} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{12 \,{\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(15*(a*b*c*d - a^2*d^2 + (b^2*c*d - a*b*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)) - 4*(2*b^2*d^2*x^4 - 3*b^2*c^2 + 20*a*b*c*d - 15*a^2*d^2 + 2
*(7*b^2*c*d - 5*a*b*d^2)*x^2)*sqrt(d*x^2 + c))/(b^4*x^2 + a*b^3), -1/12*(15*(a*b
*c*d - a^2*d^2 + (b^2*c*d - a*b*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*(2*b^2*d^2*x^4 -
3*b^2*c^2 + 20*a*b*c*d - 15*a^2*d^2 + 2*(7*b^2*c*d - 5*a*b*d^2)*x^2)*sqrt(d*x^2
+ c))/(b^4*x^2 + a*b^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227573, size = 257, normalized size = 2.04 \[ \frac{1}{6} \, d{\left (\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{3}} - \frac{3 \,{\left (\sqrt{d x^{2} + c} b^{2} c^{2} - 2 \, \sqrt{d x^{2} + c} a b c d + \sqrt{d x^{2} + c} a^{2} d^{2}\right )}}{{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} b^{3}} + \frac{2 \,{\left ({\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{4} + 6 \, \sqrt{d x^{2} + c} b^{4} c - 6 \, \sqrt{d x^{2} + c} a b^{3} d\right )}}{b^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*d*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c +
a*b*d))/(sqrt(-b^2*c + a*b*d)*b^3) - 3*(sqrt(d*x^2 + c)*b^2*c^2 - 2*sqrt(d*x^2
+ c)*a*b*c*d + sqrt(d*x^2 + c)*a^2*d^2)/(((d*x^2 + c)*b - b*c + a*d)*b^3) + 2*((
d*x^2 + c)^(3/2)*b^4 + 6*sqrt(d*x^2 + c)*b^4*c - 6*sqrt(d*x^2 + c)*a*b^3*d)/b^6)