3.541 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=222 \[ \frac{5 c^{3/2} \left (a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^6}+\frac{5 c^2 d \left (3 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^6 \sqrt{a e^2+c d^2}}-\frac{5 c \sqrt{a+c x^2} \left (a e^2+4 c d^2+2 c d e x\right )}{2 e^5 (d+e x)}+\frac{5 c \left (a+c x^2\right )^{3/2} (2 d+e x)}{6 e^3 (d+e x)^2}-\frac{\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3} \]
[Out]
(-5*c*(4*c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a + c*x^2])/(2*e^5*(d + e*x)) + (5*c*(2
*d + e*x)*(a + c*x^2)^(3/2))/(6*e^3*(d + e*x)^2) - (a + c*x^2)^(5/2)/(3*e*(d + e
*x)^3) + (5*c^(3/2)*(4*c*d^2 + a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*e
^6) + (5*c^2*d*(4*c*d^2 + 3*a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sq
rt[a + c*x^2])])/(2*e^6*Sqrt[c*d^2 + a*e^2])
_______________________________________________________________________________________
Rubi [A] time = 0.621174, antiderivative size = 222, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316
\[ \frac{5 c^{3/2} \left (a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^6}+\frac{5 c^2 d \left (3 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^6 \sqrt{a e^2+c d^2}}-\frac{5 c \sqrt{a+c x^2} \left (a e^2+4 c d^2+2 c d e x\right )}{2 e^5 (d+e x)}+\frac{5 c \left (a+c x^2\right )^{3/2} (2 d+e x)}{6 e^3 (d+e x)^2}-\frac{\left (a+c x^2\right )^{5/2}}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
(-5*c*(4*c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a + c*x^2])/(2*e^5*(d + e*x)) + (5*c*(2
*d + e*x)*(a + c*x^2)^(3/2))/(6*e^3*(d + e*x)^2) - (a + c*x^2)^(5/2)/(3*e*(d + e
*x)^3) + (5*c^(3/2)*(4*c*d^2 + a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*e
^6) + (5*c^2*d*(4*c*d^2 + 3*a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sq
rt[a + c*x^2])])/(2*e^6*Sqrt[c*d^2 + a*e^2])
_______________________________________________________________________________________
Rubi in Sympy [A] time = 62.1096, size = 214, normalized size = 0.96 \[ \frac{5 c^{\frac{3}{2}} \left (a e^{2} + 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 e^{6}} + \frac{5 c^{2} d \left (3 a e^{2} + 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 e^{6} \sqrt{a e^{2} + c d^{2}}} + \frac{5 c \left (a + c x^{2}\right )^{\frac{3}{2}} \left (4 d + 2 e x\right )}{12 e^{3} \left (d + e x\right )^{2}} - \frac{5 c \sqrt{a + c x^{2}} \left (4 a e^{2} + 16 c d^{2} + 8 c d e x\right )}{8 e^{5} \left (d + e x\right )} - \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{3 e \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**4,x)
[Out]
5*c**(3/2)*(a*e**2 + 4*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(2*e**6) + 5*c*
*2*d*(3*a*e**2 + 4*c*d**2)*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c
*d**2)))/(2*e**6*sqrt(a*e**2 + c*d**2)) + 5*c*(a + c*x**2)**(3/2)*(4*d + 2*e*x)/
(12*e**3*(d + e*x)**2) - 5*c*sqrt(a + c*x**2)*(4*a*e**2 + 16*c*d**2 + 8*c*d*e*x)
/(8*e**5*(d + e*x)) - (a + c*x**2)**(5/2)/(3*e*(d + e*x)**3)
_______________________________________________________________________________________
Mathematica [A] time = 0.454155, size = 260, normalized size = 1.17 \[ \frac{-\frac{e \sqrt{a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+15 d e x+14 e^2 x^2\right )+c^2 \left (60 d^4+150 d^3 e x+110 d^2 e^2 x^2+15 d e^3 x^3-3 e^4 x^4\right )\right )}{(d+e x)^3}+15 c^{3/2} \left (a e^2+4 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\frac{15 c^2 d \left (3 a e^2+4 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}-\frac{15 c^2 d \left (3 a e^2+4 c d^2\right ) \log (d+e x)}{\sqrt{a e^2+c d^2}}}{6 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
(-((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 15*d*e*x + 14*e^2*x^2) + c^2
*(60*d^4 + 150*d^3*e*x + 110*d^2*e^2*x^2 + 15*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x)
^3) - (15*c^2*d*(4*c*d^2 + 3*a*e^2)*Log[d + e*x])/Sqrt[c*d^2 + a*e^2] + 15*c^(3/
2)*(4*c*d^2 + a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^2*d*(4*c*d^2 + 3
*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e
^2])/(6*e^6)
_______________________________________________________________________________________
Maple [B] time = 0.023, size = 3789, normalized size = 17.1 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^(5/2)/(e*x+d)^4,x)
[Out]
5/2/(a*e^2+c*d^2)^2*c^(3/2)*a^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))-15/e^3*c^3*d^3/(a*e^2+c*d^2)^2*(c*(d/e+x)^2
-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a+1/6/e*c*d/(a*e^2+c*d^2)^2/(d/e+x)^2*
(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(7/2)+5/8/e^2*c^4*d^4/(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)*x-5/6/e*c^3*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)^(3/2)*a+25/4/e^4*c^(
9/2)*d^6/(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+75/4/e^4*c^(7/2)*d^4/(a*e^2+c*d^2)^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))*a+5/8*
c^3*d^2/(a*e^2+c*d^2)^3*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*
x+15/16*c^3*d^2/(a*e^2+c*d^2)^3*a^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e
^2)^(1/2)*x-4/3/(a*e^2+c*d^2)^2*c/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*
d^2)/e^2)^(7/2)+4/3/(a*e^2+c*d^2)^2*c^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^
2)/e^2)^(5/2)*x-1/3/e^2/(a*e^2+c*d^2)/(d/e+x)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*
e^2+c*d^2)/e^2)^(7/2)+45/2/e^5*c^4*d^5/(a*e^2+c*d^2)^2/((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+15/2/e^5*c^5*d^7/(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+35/16/e^2*c^4*d^4/(a*e^2+c*d^2)^3*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+
c*d^2)/e^2)^(1/2)*x+5/2/e*c^3*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^3+15/2/e^3*c^4*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
))*a^2+15/2/e*c^2*d/(a*e^2+c*d^2)^2/((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+105/16/e^2*c^3*d^2/(a*e^2+c*d^2)^2*a*(c*(d
/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+5/3/(a*e^2+c*d^2)^2*c^2*a*(c*
(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x+5/2/(a*e^2+c*d^2)^2*c^2*a^2
*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+1/2*c^3*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)^(5/2)*x-1/2*c^2*d^2/(a*e^2+
c*d^2)^3/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(7/2)-3/2/e*c^2
*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-1/2/e*c
^3*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)^(5/2)-5/6
/e^3*c^4*d^5/(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)+225/16/e^2*c^(5/2)*d^2/(a*e^2+c*d^2)^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))*a^2+75/16/e^2*c^(7/2)*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))*a^2-5/e^3*c^4*d^5/(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-5/2/e*c^2*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)
+(a*e^2+c*d^2)/e^2)^(3/2)*a-15/2/e*c^2*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d
/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a^2+15/2/e^7*c^5*d^7/(a*e^2+c*d^2)^2/((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))+5/4/e^4*c^5*d^
6/(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-5/2/e*
c^3*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)*a^
2+5/2/e^7*c^6*d^9/(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))+15/8/e^2*c^3*d^2/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*
c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x+15/4/e^4*c^4*d^4/(a*e^2+c*d^2)^2*(c*(d/
e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+45/2/e^3*c^3*d^3/(a*e^2+c*d^2)
^2/((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+15/16*c^(5/2)*d^2/(a*e^2+c*d^2)^3*a^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))-5/2/e^5*c^5*d^7/(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)-5/2/e^3*c^3*d^3/(a*e^2+c*d
^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-15/2/e^5*c^4*d^5/(a*
e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+5/2/e^6*c^(11
/2)*d^8/(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))+15/2/e^6*c^(9/2)*d^6/(a*e^2+c*d^2)^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))
_______________________________________________________________________________________
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)^(5/2)/(e*x + d)^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 3.64515, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)/(e*x + d)^4,x, algorithm="fricas")
[Out]
[1/12*(15*(4*c^2*d^5 + a*c*d^3*e^2 + (4*c^2*d^2*e^3 + a*c*e^5)*x^3 + 3*(4*c^2*d^
3*e^2 + a*c*d*e^4)*x^2 + 3*(4*c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(c*d^2 + a*e^2)*sq
rt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(3*c^2*e^5*x^4 - 15*c^
2*d*e^4*x^3 - 60*c^2*d^4*e - 5*a*c*d^2*e^3 - 2*a^2*e^5 - 2*(55*c^2*d^2*e^3 + 7*a
*c*e^5)*x^2 - 15*(10*c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2
+ a) + 15*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3
*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*c^3*d^5*e + 3*a*c^2*d^3*e^3)*x)*lo
g(((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)))/((e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*sqrt(c*
d^2 + a*e^2)), 1/12*(15*(4*c^2*d^5 + a*c*d^3*e^2 + (4*c^2*d^2*e^3 + a*c*e^5)*x^3
+ 3*(4*c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 3*(4*c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c*
d^2 - a*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(3*c^2*
e^5*x^4 - 15*c^2*d*e^4*x^3 - 60*c^2*d^4*e - 5*a*c*d^2*e^3 - 2*a^2*e^5 - 2*(55*c^
2*d^2*e^3 + 7*a*c*e^5)*x^2 - 15*(10*c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(-c*d^2 - a*
e^2)*sqrt(c*x^2 + a) - 30*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^
2*d*e^5)*x^3 + 3*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*c^3*d^5*e + 3*a*c^
2*d^3*e^3)*x)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*
x^2 + a))))/((e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*sqrt(-c*d^2 - a*e^2
)), 1/12*(30*(4*c^2*d^5 + a*c*d^3*e^2 + (4*c^2*d^2*e^3 + a*c*e^5)*x^3 + 3*(4*c^2
*d^3*e^2 + a*c*d*e^4)*x^2 + 3*(4*c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(c*d^2 + a*e^2)
*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) + 2*(3*c^2*e^5*x^4 - 15*c^2*d*e
^4*x^3 - 60*c^2*d^4*e - 5*a*c*d^2*e^3 - 2*a^2*e^5 - 2*(55*c^2*d^2*e^3 + 7*a*c*e^
5)*x^2 - 15*(10*c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)
+ 15*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(4*c
^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*c^3*d^5*e + 3*a*c^2*d^3*e^3)*x)*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)))/((e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*sqrt(c*d^2 +
a*e^2)), 1/6*(15*(4*c^2*d^5 + a*c*d^3*e^2 + (4*c^2*d^2*e^3 + a*c*e^5)*x^3 + 3*(
4*c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 3*(4*c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(-c*d^2 -
a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) + (3*c^2*e^5*x^4 - 15*c^2
*d*e^4*x^3 - 60*c^2*d^4*e - 5*a*c*d^2*e^3 - 2*a^2*e^5 - 2*(55*c^2*d^2*e^3 + 7*a*
c*e^5)*x^2 - 15*(10*c^2*d^3*e^2 + a*c*d*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2
+ a) - 15*(4*c^3*d^6 + 3*a*c^2*d^4*e^2 + (4*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3
*(4*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(4*c^3*d^5*e + 3*a*c^2*d^3*e^3)*x)*ar
ctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((e^
9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*sqrt(-c*d^2 - a*e^2))]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**(5/2)/(e*x+d)**4,x)
[Out]
Integral((a + c*x**2)**(5/2)/(d + e*x)**4, x)
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.616941, 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)^(5/2)/(e*x + d)^4,x, algorithm="giac")
[Out]
sage0*x