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

Optimal. Leaf size=213 \[ -\frac{5 c^{3/2} d \left (3 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^6}-\frac{5 c \sqrt{a e^2+c d^2} \left (a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^6}+\frac{5 c \sqrt{a+c x^2} \left (a e^2+4 c d^2-2 c d e x\right )}{2 e^5}+\frac{5 c \left (a+c x^2\right )^{3/2} (4 d+e x)}{6 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2} \]

[Out]

(5*c*(4*c*d^2 + a*e^2 - 2*c*d*e*x)*Sqrt[a + c*x^2])/(2*e^5) + (5*c*(4*d + e*x)*(
a + c*x^2)^(3/2))/(6*e^3*(d + e*x)) - (a + c*x^2)^(5/2)/(2*e*(d + e*x)^2) - (5*c
^(3/2)*d*(4*c*d^2 + 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*e^6) - (5*
c*Sqrt[c*d^2 + a*e^2]*(4*c*d^2 + a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^
2]*Sqrt[a + c*x^2])])/(2*e^6)

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Rubi [A]  time = 0.650509, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{5 c^{3/2} d \left (3 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^6}-\frac{5 c \sqrt{a e^2+c d^2} \left (a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^6}+\frac{5 c \sqrt{a+c x^2} \left (a e^2+4 c d^2-2 c d e x\right )}{2 e^5}+\frac{5 c \left (a+c x^2\right )^{3/2} (4 d+e x)}{6 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{5/2}}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(5*c*(4*c*d^2 + a*e^2 - 2*c*d*e*x)*Sqrt[a + c*x^2])/(2*e^5) + (5*c*(4*d + e*x)*(
a + c*x^2)^(3/2))/(6*e^3*(d + e*x)) - (a + c*x^2)^(5/2)/(2*e*(d + e*x)^2) - (5*c
^(3/2)*d*(4*c*d^2 + 3*a*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*e^6) - (5*
c*Sqrt[c*d^2 + a*e^2]*(4*c*d^2 + a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^
2]*Sqrt[a + c*x^2])])/(2*e^6)

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Rubi in Sympy [A]  time = 67.9851, size = 204, normalized size = 0.96 \[ - \frac{5 c^{\frac{3}{2}} d \left (3 a e^{2} + 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 e^{6}} + \frac{5 c \left (a + c x^{2}\right )^{\frac{3}{2}} \left (4 d + e x\right )}{6 e^{3} \left (d + e x\right )} + \frac{5 c \sqrt{a + c x^{2}} \left (4 a e^{2} + 16 c d^{2} - 8 c d e x\right )}{8 e^{5}} - \frac{5 c \sqrt{a e^{2} + c d^{2}} \left (a e^{2} + 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 e^{6}} - \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{2 e \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*c**(3/2)*d*(3*a*e**2 + 4*c*d**2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(2*e**6) +
5*c*(a + c*x**2)**(3/2)*(4*d + e*x)/(6*e**3*(d + e*x)) + 5*c*sqrt(a + c*x**2)*(
4*a*e**2 + 16*c*d**2 - 8*c*d*e*x)/(8*e**5) - 5*c*sqrt(a*e**2 + c*d**2)*(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
) - (a + c*x**2)**(5/2)/(2*e*(d + e*x)**2)

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Mathematica [A]  time = 0.43487, size = 281, normalized size = 1.32 \[ \frac{-\frac{15 c \left (a^2 e^4+5 a c d^2 e^2+4 c^2 d^4\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}+\frac{15 c \left (a^2 e^4+5 a c d^2 e^2+4 c^2 d^4\right ) \log (d+e x)}{\sqrt{a e^2+c d^2}}+\frac{e \sqrt{a+c x^2} \left (-3 a^2 e^4+a c e^2 \left (35 d^2+55 d e x+14 e^2 x^2\right )+c^2 \left (60 d^4+90 d^3 e x+20 d^2 e^2 x^2-5 d e^3 x^3+2 e^4 x^4\right )\right )}{(d+e x)^2}-15 c^{3/2} d \left (3 a e^2+4 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 e^6} \]

Antiderivative was successfully verified.

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

[Out]

((e*Sqrt[a + c*x^2]*(-3*a^2*e^4 + a*c*e^2*(35*d^2 + 55*d*e*x + 14*e^2*x^2) + c^2
*(60*d^4 + 90*d^3*e*x + 20*d^2*e^2*x^2 - 5*d*e^3*x^3 + 2*e^4*x^4)))/(d + e*x)^2
+ (15*c*(4*c^2*d^4 + 5*a*c*d^2*e^2 + a^2*e^4)*Log[d + e*x])/Sqrt[c*d^2 + a*e^2]
- 15*c^(3/2)*d*(4*c*d^2 + 3*a*e^2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - (15*c*(4
*c^2*d^4 + 5*a*c*d^2*e^2 + a^2*e^4)*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)

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Maple [B]  time = 0.02, size = 3342, normalized size = 15.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-225/16/e^2*c^(5/2)*d^3/(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^2-15/4/e^4*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)*x-75/4/e^4*c^(7/2)*d^5/(
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-45/16*c^2*d/(a*e^2+c*d^2)^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-15/8*c^2*d/(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)^(3/2)*x+15/2/e*c^2*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)^(1/2)*a^2-5/8/e^2/(a*e^2+c*d^2)*c^2*d*(c*(d/e+
x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x-5/4/e^4/(a*e^2+c*d^2)*c^3*d^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/e^3/(a*e^2+c*d^2)*c^2*(c
*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a*d^2-5/2/e/(a*e^2+c*d^2)*c/
((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^
3-5/2/e^7/(a*e^2+c*d^2)*c^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))*d^6+15/e^3*c^3*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)*a-15/2/e^7*c^5*d^8/(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))-15/8/e^2*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)*x+5/2
/e*c^2*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)
*a-75/16/e^2/(a*e^2+c*d^2)*c^(3/2)*d*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^2-25/4/e^4/(a*e^2+c*d^2)*c^(5/2)*d^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-15/2/e^6*c^(9/2)*d^7/(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))-3/2*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)*x+3/2*c*d/(a*e^2+c*d^2)^2/(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^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)^(5/2)+5/2/e/(a*e^2+c*d^2
)*c*(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^5/(a*e^2+c*d
^2)*c^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*d^4-105/16/e^2*c^3
*d^3/(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-1
5/2/e*c^2*d^2/(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^3-45/2/e^5*c^4*d^6/(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-35/16/e^2/(a*e^
2+c*d^2)*c^2*d*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-15/2/e^
3/(a*e^2+c*d^2)*c^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*d^2-15/2/e^5/(a*e^2+c*d^2)*c^3/((a*e^2+c*d^2)/e^2)^(1/2)*l
n((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*d^4-45/2/e^3*c^3*d^4/(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+1/2/e/(a*e^2+c*d^2)*c*(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/(a*e^2+c*d^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)+5/2/e^3*c^3*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)^(3/2)+15/2/e^5*c^4*d^6/(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/(a*e^2+c*d^2)*c^(7/2)*d^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))+5/6/e/(a*e^2+c*d
^2)*c*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*a+5/6/e^3/(a*e^2+c*d
^2)*c^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*d^2-45/16*c^(3/2)*
d/(a*e^2+c*d^2)^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))

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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((c*x^2 + a)^(5/2)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51082, 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)^3,x, algorithm="fricas")

[Out]

[1/12*(15*(4*c^2*d^5 + 3*a*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*a*c*d*e^4)*x^2 + 2*(4*
c^2*d^4*e + 3*a*c*d^2*e^3)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x
- a) + 15*(4*c^2*d^4 + a*c*d^2*e^2 + (4*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(4*c^2*d
^3*e + a*c*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*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 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a
))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(2*c^2*e^5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*d^4*
e + 35*a*c*d^2*e^3 - 3*a^2*e^5 + 2*(10*c^2*d^2*e^3 + 7*a*c*e^5)*x^2 + 5*(18*c^2*
d^3*e^2 + 11*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6), -1/
12*(30*(4*c^2*d^5 + 3*a*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*a*c*d*e^4)*x^2 + 2*(4*c^2
*d^4*e + 3*a*c*d^2*e^3)*x)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - 15*
(4*c^2*d^4 + a*c*d^2*e^2 + (4*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(4*c^2*d^3*e + a*c*
d*e^3)*x)*sqrt(c*d^2 + a*e^2)*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 - 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2
+ 2*d*e*x + d^2)) - 2*(2*c^2*e^5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*d^4*e + 35*a*c*
d^2*e^3 - 3*a^2*e^5 + 2*(10*c^2*d^2*e^3 + 7*a*c*e^5)*x^2 + 5*(18*c^2*d^3*e^2 + 1
1*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6), 1/12*(30*(4*c^
2*d^4 + a*c*d^2*e^2 + (4*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(4*c^2*d^3*e + a*c*d*e^3
)*x)*sqrt(-c*d^2 - a*e^2)*arctan((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2
+ a))) + 15*(4*c^2*d^5 + 3*a*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*a*c*d*e^4)*x^2 + 2*(
4*c^2*d^4*e + 3*a*c*d^2*e^3)*x)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)
*x - a) + 2*(2*c^2*e^5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*d^4*e + 35*a*c*d^2*e^3 - 3
*a^2*e^5 + 2*(10*c^2*d^2*e^3 + 7*a*c*e^5)*x^2 + 5*(18*c^2*d^3*e^2 + 11*a*c*d*e^4
)*x)*sqrt(c*x^2 + a))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6), -1/6*(15*(4*c^2*d^5 + 3*a
*c*d^3*e^2 + (4*c^2*d^3*e^2 + 3*a*c*d*e^4)*x^2 + 2*(4*c^2*d^4*e + 3*a*c*d^2*e^3)
*x)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - 15*(4*c^2*d^4 + a*c*d^2*e^
2 + (4*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(4*c^2*d^3*e + a*c*d*e^3)*x)*sqrt(-c*d^2 -
a*e^2)*arctan((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) - (2*c^2*e^
5*x^4 - 5*c^2*d*e^4*x^3 + 60*c^2*d^4*e + 35*a*c*d^2*e^3 - 3*a^2*e^5 + 2*(10*c^2*
d^2*e^3 + 7*a*c*e^5)*x^2 + 5*(18*c^2*d^3*e^2 + 11*a*c*d*e^4)*x)*sqrt(c*x^2 + a))
/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.575304, 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)^3,x, algorithm="giac")

[Out]

sage0*x