3.383 \(\int \frac{x^5}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=134 \[ -\frac{d \left (a+b x^2\right )^{3/2}}{3 b^3 c^2}+\frac{\left (a c^2-d^2\right )^2 \log \left (c \sqrt{a+b x^2}+d\right )}{b^3 c^5}+\frac{d \sqrt{a+b x^2} \left (2 a c^2-d^2\right )}{b^3 c^4}+\frac{\left (a+b x^2\right )^2}{4 b^3 c}-\frac{x^2 \left (2 a c^2-d^2\right )}{2 b^2 c^3} \]

[Out]

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

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Rubi [A]  time = 0.619294, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{d \left (a+b x^2\right )^{3/2}}{3 b^3 c^2}+\frac{\left (a c^2-d^2\right )^2 \log \left (c \sqrt{a+b x^2}+d\right )}{b^3 c^5}+\frac{d \sqrt{a+b x^2} \left (2 a c^2-d^2\right )}{b^3 c^4}+\frac{\left (a+b x^2\right )^2}{4 b^3 c}-\frac{x^2 \left (2 a c^2-d^2\right )}{2 b^2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\left (a + b x^{2}\right )^{2}}{4 b^{3} c} - \frac{d \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{3} c^{2}} + \frac{\left (- 2 a c^{2} + d^{2}\right ) \int ^{\sqrt{a + b x^{2}}} x\, dx}{b^{3} c^{3}} - \frac{\left (- 2 a c^{2} + d^{2}\right ) \int ^{\sqrt{a + b x^{2}}} d\, dx}{b^{3} c^{4}} + \frac{\left (- a c^{2} + d^{2}\right )^{2} \log{\left (c \sqrt{a + b x^{2}} + d \right )}}{b^{3} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**2/(4*b**3*c) - d*(a + b*x**2)**(3/2)/(3*b**3*c**2) + (-2*a*c**2 +
d**2)*Integral(x, (x, sqrt(a + b*x**2)))/(b**3*c**3) - (-2*a*c**2 + d**2)*Integr
al(d, (x, sqrt(a + b*x**2)))/(b**3*c**4) + (-a*c**2 + d**2)**2*log(c*sqrt(a + b*
x**2) + d)/(b**3*c**5)

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Mathematica [A]  time = 0.335107, size = 161, normalized size = 1.2 \[ \frac{c \left (a \left (20 c^2 d \sqrt{a+b x^2}-6 b c^3 x^2\right )+2 b c d x^2 \left (3 d-2 c \sqrt{a+b x^2}\right )-12 d^3 \sqrt{a+b x^2}+3 b^2 c^3 x^4\right )+6 \left (d^2-a c^2\right )^2 \log \left (a c^2+b c^2 x^2-d^2\right )+12 \left (d^2-a c^2\right )^2 \tanh ^{-1}\left (\frac{c \sqrt{a+b x^2}}{d}\right )}{12 b^3 c^5} \]

Antiderivative was successfully verified.

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

[Out]

(c*(3*b^2*c^3*x^4 - 12*d^3*Sqrt[a + b*x^2] + 2*b*c*d*x^2*(3*d - 2*c*Sqrt[a + b*x
^2]) + a*(-6*b*c^3*x^2 + 20*c^2*d*Sqrt[a + b*x^2])) + 12*(-(a*c^2) + d^2)^2*ArcT
anh[(c*Sqrt[a + b*x^2])/d] + 6*(-(a*c^2) + d^2)^2*Log[a*c^2 - d^2 + b*c^2*x^2])/
(12*b^3*c^5)

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Maple [B]  time = 0.086, size = 4947, normalized size = 36.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a
*c^2-d^2))^(1/2))/c^2*d^5/(d^2/c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(
1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c
^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2
))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))*a+1/4/b/c*
x^4+1/2*a^2/c/b^3*ln(b*c^2*x^2+a*c^2-d^2)+1/2/b^2/c^3*x^2*d^2+1/2/b^3/c^5*d^4*ln
(b*c^2*x^2+a*c^2-d^2)+1/b^(5/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((
-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/c^2*(-c^2*b*(a*c^2-d^2))^(1/2)*ln((-
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+(
(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c
^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))*a*d^3+1/b^2/((-a*b)^(1/2)*c^2+(-c
^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/c^2*d^5/(
d^2/c^2)^(1/2)*ln((2*d^2/c^2-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-
d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2
*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1
/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))*a-1/2*d/b^(5/2)*c^2*a^2/((-a*b)^(1/2)
*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*(
-a*b)^(1/2)*ln(((x-1/b*(-a*b)^(1/2))*b+(-a*b)^(1/2))/b^(1/2)+((x-1/b*(-a*b)^(1/2
))^2*b+2*(-a*b)^(1/2)*(x-1/b*(-a*b)^(1/2)))^(1/2))+1/2*d/b^(5/2)*c^2*a^2/((-a*b)
^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1
/2))*(-a*b)^(1/2)*ln(((x+1/b*(-a*b)^(1/2))*b-(-a*b)^(1/2))/b^(1/2)+((x+1/b*(-a*b
)^(1/2))^2*b-2*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2))-1/b^(5/2)/((-a*b)^(1/2)
*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/c
^2*(-c^2*b*(a*c^2-d^2))^(1/2)*ln(((-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x-(-c^2*b*(a*c
^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c
^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))
*a*d^3+1/2/b^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(
-c^2*b*(a*c^2-d^2))^(1/2))/c^2*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2
*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)*d^
5+1/2*d/b^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^
2*b*(a*c^2-d^2))^(1/2))*c^2*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*
(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)*a^2+1
/2*d/b^(5/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c
^2*b*(a*c^2-d^2))^(1/2))*(-c^2*b*(a*c^2-d^2))^(1/2)*ln(((-c^2*b*(a*c^2-d^2))^(1/
2)/c^2+(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x-(-c^2*b*(a*c^2-d^2))^
(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/
c^2/b)+d^2/c^2)^(1/2))*a^2+1/2/b^(5/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1
/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/c^4*(-c^2*b*(a*c^2-d^2))^(1/2
)*ln(((-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(
1/2)+((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*
(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))*d^5-1/2/b^2/((-a*b)^(1/2)*c
^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*d^3
/(d^2/c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^
2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b
+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^
(1/2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))*a^2-1/2/b^2/((-a*b)^(1/2)*c^2+(-c^2
*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/c^4*d^7/(d^
2/c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^
2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(
-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2
))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))+1/2*d/b^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*
c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*c^2*((x+(-c^2*b*(
a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-
d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)*a^2-1/2*d/b^(5/2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a
*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*(-c^2*b*(a*c^2-d
^2))^(1/2)*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2
/b)*b)/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^
(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))*a^2-1/2/b^(5/2)/(
(-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^
2))^(1/2))/c^4*(-c^2*b*(a*c^2-d^2))^(1/2)*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x
+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2
/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^
2/c^2)^(1/2))*d^5-1/2/b^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^
(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*d^3/(d^2/c^2)^(1/2)*ln((2*d^2/c^2-2*(-c^2*
b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*
((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-
c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^
2/b))*a^2-1/2/b^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^
2-(-c^2*b*(a*c^2-d^2))^(1/2))/c^4*d^7/(d^2/c^2)^(1/2)*ln((2*d^2/c^2-2*(-c^2*b*(a
*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x+
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*
b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)
)-1/2*d/b^2*c^2*a^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*
c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x-1/b*(-a*b)^(1/2))^2*b+2*(-a*b)^(1/2)*(x-1/b*
(-a*b)^(1/2)))^(1/2)-1/2*d/b^2*c^2*a^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1
/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x+1/b*(-a*b)^(1/2))^2*b-2*(
-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2)-1/2*a/c/b^2*x^2-1/b^2/((-a*b)^(1/2)*c^2+
(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x-(-
c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*
(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)*a*d^3+1/2/b^2/((-a*b)^(1/2)*c^2+(-c^2*b
*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/c^2*((x-(-c^2
*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*
c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)*d^5-1/b^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2
-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x+(-c^2*b*(a*c^2-d
^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(
1/2)/c^2/b)+d^2/c^2)^(1/2)*a*d^3-a/c^3/b^3*d^2*ln(b*c^2*x^2+a*c^2-d^2)-1/3*d*(b*
x^2+a)^(3/2)/b^3/c^2

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Maxima [A]  time = 0.707384, size = 169, normalized size = 1.26 \[ \frac{\frac{3 \,{\left (b x^{2} + a\right )}^{2} c^{3} - 4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} c^{2} d - 6 \,{\left (2 \, a c^{3} - c d^{2}\right )}{\left (b x^{2} + a\right )} + 12 \,{\left (2 \, a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a}}{c^{4}} + \frac{12 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt{b x^{2} + a} c + d\right )}{c^{5}}}{12 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((3*(b*x^2 + a)^2*c^3 - 4*(b*x^2 + a)^(3/2)*c^2*d - 6*(2*a*c^3 - c*d^2)*(b*
x^2 + a) + 12*(2*a*c^2*d - d^3)*sqrt(b*x^2 + a))/c^4 + 12*(a^2*c^4 - 2*a*c^2*d^2
+ d^4)*log(sqrt(b*x^2 + a)*c + d)/c^5)/b^3

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Fricas [A]  time = 0.353589, size = 315, normalized size = 2.35 \[ \frac{3 \, b^{2} c^{4} x^{4} - 6 \,{\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2} + 6 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) + 3 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - 3 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - 4 \,{\left (b c^{3} d x^{2} - 5 \, a c^{3} d + 3 \, c d^{3}\right )} \sqrt{b x^{2} + a}}{12 \, b^{3} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*b^2*c^4*x^4 - 6*(a*b*c^4 - b*c^2*d^2)*x^2 + 6*(a^2*c^4 - 2*a*c^2*d^2 + d
^4)*log(b*c^2*x^2 + a*c^2 - d^2) + 3*(a^2*c^4 - 2*a*c^2*d^2 + d^4)*log(-(b*c^2*x
^2 + a*c^2 + 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) - 3*(a^2*c^4 - 2*a*c^2*d^2 + d^4)
*log(-(b*c^2*x^2 + a*c^2 - 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) - 4*(b*c^3*d*x^2 -
5*a*c^3*d + 3*c*d^3)*sqrt(b*x^2 + a))/(b^3*c^5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277159, size = 209, normalized size = 1.56 \[ \frac{{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )}{\rm ln}\left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{b^{3} c^{5}} + \frac{3 \,{\left (b x^{2} + a\right )}^{2} b^{9} c^{3} - 12 \,{\left (b x^{2} + a\right )} a b^{9} c^{3} - 4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{9} c^{2} d + 24 \, \sqrt{b x^{2} + a} a b^{9} c^{2} d + 6 \,{\left (b x^{2} + a\right )} b^{9} c d^{2} - 12 \, \sqrt{b x^{2} + a} b^{9} d^{3}}{12 \, b^{12} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^2*c^4 - 2*a*c^2*d^2 + d^4)*ln(abs(sqrt(b*x^2 + a)*c + d))/(b^3*c^5) + 1/12*(3
*(b*x^2 + a)^2*b^9*c^3 - 12*(b*x^2 + a)*a*b^9*c^3 - 4*(b*x^2 + a)^(3/2)*b^9*c^2*
d + 24*sqrt(b*x^2 + a)*a*b^9*c^2*d + 6*(b*x^2 + a)*b^9*c*d^2 - 12*sqrt(b*x^2 + a
)*b^9*d^3)/(b^12*c^4)