3.532 \(\int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=195 \[ -\frac{3 a^2 c^3 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{3 a c^2 d \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )} \]

[Out]

(-3*a*c^2*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (
c*d*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(4*(c*d^2 + a*e^2)^2*(d + e*x)^4) - (e*(a +
c*x^2)^(5/2))/(5*(c*d^2 + a*e^2)*(d + e*x)^5) - (3*a^2*c^3*d*ArcTanh[(a*e - c*d
*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.342111, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3 a^2 c^3 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{3 a c^2 d \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(-3*a*c^2*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (
c*d*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(4*(c*d^2 + a*e^2)^2*(d + e*x)^4) - (e*(a +
c*x^2)^(5/2))/(5*(c*d^2 + a*e^2)*(d + e*x)^5) - (3*a^2*c^3*d*ArcTanh[(a*e - c*d
*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.3018, size = 187, normalized size = 0.96 \[ - \frac{3 a^{2} c^{3} d \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{8 \left (a e^{2} + c d^{2}\right )^{\frac{7}{2}}} - \frac{3 a c^{2} d \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right )}{16 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{3}} - \frac{c d \left (a + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - 2 c d x\right )}{8 \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{e \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 \left (d + e x\right )^{5} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**6,x)

[Out]

-3*a**2*c**3*d*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(8*
(a*e**2 + c*d**2)**(7/2)) - 3*a*c**2*d*sqrt(a + c*x**2)*(2*a*e - 2*c*d*x)/(16*(d
+ e*x)**2*(a*e**2 + c*d**2)**3) - c*d*(a + c*x**2)**(3/2)*(2*a*e - 2*c*d*x)/(8*
(d + e*x)**4*(a*e**2 + c*d**2)**2) - e*(a + c*x**2)**(5/2)/(5*(d + e*x)**5*(a*e*
*2 + c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.664571, size = 272, normalized size = 1.39 \[ \frac{1}{40} \left (-\frac{15 a^2 c^3 d \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac{15 a^2 c^3 d \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}}+\frac{\sqrt{a+c x^2} \left (-8 a^4 e^5-2 a^3 c e^3 \left (13 d^2+5 d e x+8 e^2 x^2\right )-a^2 c^2 e \left (33 d^4+45 d^3 e x+77 d^2 e^2 x^2+25 d e^3 x^3+8 e^4 x^4\right )+a c^3 d^2 x \left (25 d^3+29 d^2 e x+45 d e^2 x^2+9 e^3 x^3\right )+2 c^4 d^4 x^3 (5 d+e x)\right )}{(d+e x)^5 \left (a e^2+c d^2\right )^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

((Sqrt[a + c*x^2]*(-8*a^4*e^5 + 2*c^4*d^4*x^3*(5*d + e*x) - 2*a^3*c*e^3*(13*d^2
+ 5*d*e*x + 8*e^2*x^2) + a*c^3*d^2*x*(25*d^3 + 29*d^2*e*x + 45*d*e^2*x^2 + 9*e^3
*x^3) - a^2*c^2*e*(33*d^4 + 45*d^3*e*x + 77*d^2*e^2*x^2 + 25*d*e^3*x^3 + 8*e^4*x
^4)))/((c*d^2 + a*e^2)^3*(d + e*x)^5) + (15*a^2*c^3*d*Log[d + e*x])/(c*d^2 + a*e
^2)^(7/2) - (15*a^2*c^3*d*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]]
)/(c*d^2 + a*e^2)^(7/2))/40

_______________________________________________________________________________________

Maple [B]  time = 0.03, size = 3622, normalized size = 18.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d)^6,x)

[Out]

-1/8*c^5*d^4/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/
2)*x-3/16*c^(9/2)*d^4/(a*e^2+c*d^2)^5*a^2*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+
x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+1/8*c^4*d^4/(a*e^2+c*d^2)^5/(d/e+
x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)+3/4/e^4*c^(11/2)*d^6/(a
*e^2+c*d^2)^4*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+
c*d^2)/e^2)^(1/2))+1/8/e*c^5*d^5/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a
*e^2+c*d^2)/e^2)^(3/2)+3/8/e^3*c^6*d^7/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e
+x)+(a*e^2+c*d^2)/e^2)^(1/2)-1/4/e*c^4*d^3/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*
(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-3/4/e^3*c^5*d^5/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*
c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8/e*c^3*d/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-
2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+3/8/e^3*c^4*d^3/(a*e^2+c*d^2)^3*(c*(d/e
+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-3/8/e^4*c^(13/2)*d^8/(a*e^2+c*d^2
)^5*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2
)^(1/2))-3/8/e^4*c^(9/2)*d^4/(a*e^2+c*d^2)^3*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d
/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))-3/8/e*c^5*d^5/(a*e^2+c*d^2)^5/
((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^
2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^
2-3/4/e^3*c^6*d^7/(a*e^2+c*d^2)^5/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/
e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*
e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a-3/8/e*c^3*d/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2
)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(
d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2+3/4/e*c^4*d^3/(a
*e^2+c*d^2)^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+
2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2
))/(d/e+x))*a^2-1/5/e^4/(a*e^2+c*d^2)/(d/e+x)^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*
e^2+c*d^2)/e^2)^(5/2)+9/16*c^(7/2)*d^2/(a*e^2+c*d^2)^4*a^2*ln((-c*d/e+c*(d/e+x))
/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))-3/8*c^3*d^2/(a*e
^2+c*d^2)^4/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-1/8/e*
c^2*d/(a*e^2+c*d^2)^3/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^
(5/2)+3/8/e*c^3*d/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2
)^(1/2)*a+3/2/e^3*c^5*d^5/(a*e^2+c*d^2)^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2
+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/
e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a-3/4/e^3*c^4*d^3/(a*e^2+c*d^2)^3/((a*e^
2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2
)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a+9/8/e^
2*c^(9/2)*d^4/(a*e^2+c*d^2)^4*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e
*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a-3/16*c^5*d^4/(a*e^2+c*d^2)^5*a*(c*(d/e+x)^2
-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+9/16*c^4*d^2/(a*e^2+c*d^2)^4*a*(c*(d
/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-1/4/e^3*c*d/(a*e^2+c*d^2)^2/(
d/e+x)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-3/16/e^2*c^4*d^2/
(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/8/e^5*
c^5*d^5/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/
e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)
/e^2)^(1/2))/(d/e+x))-3/16/e^2*c^6*d^6/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/e*(d/e
+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+3/8/e*c^5*d^5/(a*e^2+c*d^2)^5*(c*(d/e+x)^2-2*c*d/
e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a-3/8/e^5*c^7*d^9/(a*e^2+c*d^2)^5/((a*e^2+c*d
^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/
2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))+3/8/e^2*c^5*d
^4/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/4/e
*c^4*d^3/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a
-9/16/e^2*c^(7/2)*d^2/(a*e^2+c*d^2)^3*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2
-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a-9/16/e^2*c^(11/2)*d^6/(a*e^2+c*d^2)
^5*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)
^(1/2))*a+3/4/e^5*c^6*d^7/(a*e^2+c*d^2)^4/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2
+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/
e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))-1/4/e^2*c^2*d^2/(a*e^2+c*d^2)^3/(d/e+x)^
3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-1/8/e*c^3*d^3/(a*e^2+c*d
^2)^4/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)+3/8*c^4*d^
2/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x

_______________________________________________________________________________________

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((c*x^2 + a)^(3/2)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 3.11436, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^6,x, algorithm="fricas")

[Out]

[-1/80*(2*(33*a^2*c^2*d^4*e + 26*a^3*c*d^2*e^3 + 8*a^4*e^5 - (2*c^4*d^4*e + 9*a*
c^3*d^2*e^3 - 8*a^2*c^2*e^5)*x^4 - 5*(2*c^4*d^5 + 9*a*c^3*d^3*e^2 - 5*a^2*c^2*d*
e^4)*x^3 - (29*a*c^3*d^4*e - 77*a^2*c^2*d^2*e^3 - 16*a^3*c*e^5)*x^2 - 5*(5*a*c^3
*d^5 - 9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)
- 15*(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*x^4 + 10*a^2*c^3*d^3*e^3*x^3 + 10*a
^2*c^3*d^4*e^2*x^2 + 5*a^2*c^3*d^5*e*x + a^2*c^3*d^6)*log(((2*a*c*d*e*x - a*c*d^
2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2)*sqrt(c*d^2 + a*e^2) + 2*(a*c*d^2*e +
a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)))/
((c^3*d^11 + 3*a*c^2*d^9*e^2 + 3*a^2*c*d^7*e^4 + a^3*d^5*e^6 + (c^3*d^6*e^5 + 3*
a*c^2*d^4*e^7 + 3*a^2*c*d^2*e^9 + a^3*e^11)*x^5 + 5*(c^3*d^7*e^4 + 3*a*c^2*d^5*e
^6 + 3*a^2*c*d^3*e^8 + a^3*d*e^10)*x^4 + 10*(c^3*d^8*e^3 + 3*a*c^2*d^6*e^5 + 3*a
^2*c*d^4*e^7 + a^3*d^2*e^9)*x^3 + 10*(c^3*d^9*e^2 + 3*a*c^2*d^7*e^4 + 3*a^2*c*d^
5*e^6 + a^3*d^3*e^8)*x^2 + 5*(c^3*d^10*e + 3*a*c^2*d^8*e^3 + 3*a^2*c*d^6*e^5 + a
^3*d^4*e^7)*x)*sqrt(c*d^2 + a*e^2)), -1/40*((33*a^2*c^2*d^4*e + 26*a^3*c*d^2*e^3
+ 8*a^4*e^5 - (2*c^4*d^4*e + 9*a*c^3*d^2*e^3 - 8*a^2*c^2*e^5)*x^4 - 5*(2*c^4*d^
5 + 9*a*c^3*d^3*e^2 - 5*a^2*c^2*d*e^4)*x^3 - (29*a*c^3*d^4*e - 77*a^2*c^2*d^2*e^
3 - 16*a^3*c*e^5)*x^2 - 5*(5*a*c^3*d^5 - 9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*x)*s
qrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 15*(a^2*c^3*d*e^5*x^5 + 5*a^2*c^3*d^2*e^4*
x^4 + 10*a^2*c^3*d^3*e^3*x^3 + 10*a^2*c^3*d^4*e^2*x^2 + 5*a^2*c^3*d^5*e*x + a^2*
c^3*d^6)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 +
a))))/((c^3*d^11 + 3*a*c^2*d^9*e^2 + 3*a^2*c*d^7*e^4 + a^3*d^5*e^6 + (c^3*d^6*e
^5 + 3*a*c^2*d^4*e^7 + 3*a^2*c*d^2*e^9 + a^3*e^11)*x^5 + 5*(c^3*d^7*e^4 + 3*a*c^
2*d^5*e^6 + 3*a^2*c*d^3*e^8 + a^3*d*e^10)*x^4 + 10*(c^3*d^8*e^3 + 3*a*c^2*d^6*e^
5 + 3*a^2*c*d^4*e^7 + a^3*d^2*e^9)*x^3 + 10*(c^3*d^9*e^2 + 3*a*c^2*d^7*e^4 + 3*a
^2*c*d^5*e^6 + a^3*d^3*e^8)*x^2 + 5*(c^3*d^10*e + 3*a*c^2*d^8*e^3 + 3*a^2*c*d^6*
e^5 + a^3*d^4*e^7)*x)*sqrt(-c*d^2 - a*e^2))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(3/2)/(e*x+d)**6,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**6, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.598095, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x