3.543 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=314 \[ -\frac{c^2 \sqrt{a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{8 e^5 (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 e^6 \left (a e^2+c d^2\right )^{5/2}}+\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^6}-\frac{c \left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^3 (d+e x)^4 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5} \]

[Out]

-(c^2*(d*(8*c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + e*(12*c^2*d^4 + 23*a*c*d^2*e^2
+ 8*a^2*e^4)*x)*Sqrt[a + c*x^2])/(8*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (c*(d*
(4*c*d^2 + a*e^2) + e*(7*c*d^2 + 4*a*e^2)*x)*(a + c*x^2)^(3/2))/(12*e^3*(c*d^2 +
a*e^2)*(d + e*x)^4) - (a + c*x^2)^(5/2)/(5*e*(d + e*x)^5) + (c^(5/2)*ArcTanh[(S
qrt[c]*x)/Sqrt[a + c*x^2]])/e^6 + (c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^
4)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*e^6*(c*d^2 +
a*e^2)^(5/2))

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Rubi [A]  time = 0.931471, antiderivative size = 314, 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 \[ -\frac{c^2 \sqrt{a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{8 e^5 (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 e^6 \left (a e^2+c d^2\right )^{5/2}}+\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^6}-\frac{c \left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^3 (d+e x)^4 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

-(c^2*(d*(8*c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + e*(12*c^2*d^4 + 23*a*c*d^2*e^2
+ 8*a^2*e^4)*x)*Sqrt[a + c*x^2])/(8*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (c*(d*
(4*c*d^2 + a*e^2) + e*(7*c*d^2 + 4*a*e^2)*x)*(a + c*x^2)^(3/2))/(12*e^3*(c*d^2 +
a*e^2)*(d + e*x)^4) - (a + c*x^2)^(5/2)/(5*e*(d + e*x)^5) + (c^(5/2)*ArcTanh[(S
qrt[c]*x)/Sqrt[a + c*x^2]])/e^6 + (c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^
4)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*e^6*(c*d^2 +
a*e^2)^(5/2))

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Rubi in Sympy [A]  time = 100.401, size = 296, normalized size = 0.94 \[ \frac{c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{6}} + \frac{c^{3} d \left (15 a^{2} e^{4} + 20 a c d^{2} e^{2} + 8 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{8 e^{6} \left (a e^{2} + c d^{2}\right )^{\frac{5}{2}}} - \frac{c^{2} \sqrt{a + c x^{2}} \left (d \left (a^{2} e^{4} + 12 a c d^{2} e^{2} + 8 c^{2} d^{4}\right ) + e x \left (8 a^{2} e^{4} + 23 a c d^{2} e^{2} + 12 c^{2} d^{4}\right )\right )}{8 e^{5} \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{c \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d \left (a e^{2} + 4 c d^{2}\right ) + e x \left (4 a e^{2} + 7 c d^{2}\right )\right )}{12 e^{3} \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )} - \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**6,x)

[Out]

c**(5/2)*atanh(sqrt(c)*x/sqrt(a + c*x**2))/e**6 + c**3*d*(15*a**2*e**4 + 20*a*c*
d**2*e**2 + 8*c**2*d**4)*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d
**2)))/(8*e**6*(a*e**2 + c*d**2)**(5/2)) - c**2*sqrt(a + c*x**2)*(d*(a**2*e**4 +
12*a*c*d**2*e**2 + 8*c**2*d**4) + e*x*(8*a**2*e**4 + 23*a*c*d**2*e**2 + 12*c**2
*d**4))/(8*e**5*(d + e*x)**2*(a*e**2 + c*d**2)**2) - c*(a + c*x**2)**(3/2)*(d*(a
*e**2 + 4*c*d**2) + e*x*(4*a*e**2 + 7*c*d**2))/(12*e**3*(d + e*x)**4*(a*e**2 + c
*d**2)) - (a + c*x**2)**(5/2)/(5*e*(d + e*x)**5)

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Mathematica [A]  time = 0.930677, size = 359, normalized size = 1.14 \[ \frac{-\frac{e \sqrt{a+c x^2} \left (c^2 (d+e x)^4 \left (184 a^2 e^4+503 a c d^2 e^2+274 c^2 d^4\right )-c^2 d (d+e x)^3 \left (311 a e^2+326 c d^2\right ) \left (a e^2+c d^2\right )-126 c d (d+e x) \left (a e^2+c d^2\right )^3+2 c (d+e x)^2 \left (44 a e^2+137 c d^2\right ) \left (a e^2+c d^2\right )^2+24 \left (a e^2+c d^2\right )^4\right )}{(d+e x)^5 \left (a e^2+c d^2\right )^2}+\frac{15 c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{5/2}}-\frac{15 c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{5/2}}+120 c^{5/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{120 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

(-((e*Sqrt[a + c*x^2]*(24*(c*d^2 + a*e^2)^4 - 126*c*d*(c*d^2 + a*e^2)^3*(d + e*x
) + 2*c*(c*d^2 + a*e^2)^2*(137*c*d^2 + 44*a*e^2)*(d + e*x)^2 - c^2*d*(c*d^2 + a*
e^2)*(326*c*d^2 + 311*a*e^2)*(d + e*x)^3 + c^2*(274*c^2*d^4 + 503*a*c*d^2*e^2 +
184*a^2*e^4)*(d + e*x)^4))/((c*d^2 + a*e^2)^2*(d + e*x)^5)) - (15*c^3*d*(8*c^2*d
^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*Log[d + e*x])/(c*d^2 + a*e^2)^(5/2) + 120*c^(5
/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + (15*c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 +
15*a^2*e^4)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*
e^2)^(5/2))/(120*e^6)

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Maple [B]  time = 0.037, size = 5921, normalized size = 18.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(e*x+d)^6,x)

[Out]

result too large to display

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 88.568, 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)^6,x, algorithm="fricas")

[Out]

[1/240*(120*(c^4*d^9 + 2*a*c^3*d^7*e^2 + a^2*c^2*d^5*e^4 + (c^4*d^4*e^5 + 2*a*c^
3*d^2*e^7 + a^2*c^2*e^9)*x^5 + 5*(c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8)
*x^4 + 10*(c^4*d^6*e^3 + 2*a*c^3*d^4*e^5 + a^2*c^2*d^2*e^7)*x^3 + 10*(c^4*d^7*e^
2 + 2*a*c^3*d^5*e^4 + a^2*c^2*d^3*e^6)*x^2 + 5*(c^4*d^8*e + 2*a*c^3*d^6*e^3 + a^
2*c^2*d^4*e^5)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*s
qrt(c)*x - a) - 2*(120*c^4*d^8*e + 220*a*c^3*d^6*e^3 + 89*a^2*c^2*d^4*e^5 + 58*a
^3*c*d^2*e^7 + 24*a^4*e^9 + (274*c^4*d^4*e^5 + 503*a*c^3*d^2*e^7 + 184*a^2*c^2*e
^9)*x^4 + 5*(154*c^4*d^5*e^4 + 275*a*c^3*d^3*e^6 + 85*a^2*c^2*d*e^8)*x^3 + (940*
c^4*d^6*e^3 + 1743*a*c^3*d^4*e^5 + 621*a^2*c^2*d^2*e^7 + 88*a^3*c*e^9)*x^2 + 5*(
108*c^4*d^7*e^2 + 199*a*c^3*d^5*e^4 + 65*a^2*c^2*d^3*e^6 + 10*a^3*c*d*e^8)*x)*sq
rt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 15*(8*c^5*d^10 + 20*a*c^4*d^8*e^2 + 15*a^2*c
^3*d^6*e^4 + (8*c^5*d^5*e^5 + 20*a*c^4*d^3*e^7 + 15*a^2*c^3*d*e^9)*x^5 + 5*(8*c^
5*d^6*e^4 + 20*a*c^4*d^4*e^6 + 15*a^2*c^3*d^2*e^8)*x^4 + 10*(8*c^5*d^7*e^3 + 20*
a*c^4*d^5*e^5 + 15*a^2*c^3*d^3*e^7)*x^3 + 10*(8*c^5*d^8*e^2 + 20*a*c^4*d^6*e^4 +
15*a^2*c^3*d^4*e^6)*x^2 + 5*(8*c^5*d^9*e + 20*a*c^4*d^7*e^3 + 15*a^2*c^3*d^5*e^
5)*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)))/((c^2*d^9*e^6 + 2*a*c*d^7*e^8 + a^2*d^5*e^10 + (
c^2*d^4*e^11 + 2*a*c*d^2*e^13 + a^2*e^15)*x^5 + 5*(c^2*d^5*e^10 + 2*a*c*d^3*e^12
+ a^2*d*e^14)*x^4 + 10*(c^2*d^6*e^9 + 2*a*c*d^4*e^11 + a^2*d^2*e^13)*x^3 + 10*(
c^2*d^7*e^8 + 2*a*c*d^5*e^10 + a^2*d^3*e^12)*x^2 + 5*(c^2*d^8*e^7 + 2*a*c*d^6*e^
9 + a^2*d^4*e^11)*x)*sqrt(c*d^2 + a*e^2)), 1/120*(60*(c^4*d^9 + 2*a*c^3*d^7*e^2
+ a^2*c^2*d^5*e^4 + (c^4*d^4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x^5 + 5*(c^4*d
^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8)*x^4 + 10*(c^4*d^6*e^3 + 2*a*c^3*d^4*e^
5 + a^2*c^2*d^2*e^7)*x^3 + 10*(c^4*d^7*e^2 + 2*a*c^3*d^5*e^4 + a^2*c^2*d^3*e^6)*
x^2 + 5*(c^4*d^8*e + 2*a*c^3*d^6*e^3 + a^2*c^2*d^4*e^5)*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) - (120*c^4*d^8*e + 220*a
*c^3*d^6*e^3 + 89*a^2*c^2*d^4*e^5 + 58*a^3*c*d^2*e^7 + 24*a^4*e^9 + (274*c^4*d^4
*e^5 + 503*a*c^3*d^2*e^7 + 184*a^2*c^2*e^9)*x^4 + 5*(154*c^4*d^5*e^4 + 275*a*c^3
*d^3*e^6 + 85*a^2*c^2*d*e^8)*x^3 + (940*c^4*d^6*e^3 + 1743*a*c^3*d^4*e^5 + 621*a
^2*c^2*d^2*e^7 + 88*a^3*c*e^9)*x^2 + 5*(108*c^4*d^7*e^2 + 199*a*c^3*d^5*e^4 + 65
*a^2*c^2*d^3*e^6 + 10*a^3*c*d*e^8)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 15*
(8*c^5*d^10 + 20*a*c^4*d^8*e^2 + 15*a^2*c^3*d^6*e^4 + (8*c^5*d^5*e^5 + 20*a*c^4*
d^3*e^7 + 15*a^2*c^3*d*e^9)*x^5 + 5*(8*c^5*d^6*e^4 + 20*a*c^4*d^4*e^6 + 15*a^2*c
^3*d^2*e^8)*x^4 + 10*(8*c^5*d^7*e^3 + 20*a*c^4*d^5*e^5 + 15*a^2*c^3*d^3*e^7)*x^3
+ 10*(8*c^5*d^8*e^2 + 20*a*c^4*d^6*e^4 + 15*a^2*c^3*d^4*e^6)*x^2 + 5*(8*c^5*d^9
*e + 20*a*c^4*d^7*e^3 + 15*a^2*c^3*d^5*e^5)*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))))/((c^2*d^9*e^6 + 2*a*c*d^7*e^8 + a^2
*d^5*e^10 + (c^2*d^4*e^11 + 2*a*c*d^2*e^13 + a^2*e^15)*x^5 + 5*(c^2*d^5*e^10 + 2
*a*c*d^3*e^12 + a^2*d*e^14)*x^4 + 10*(c^2*d^6*e^9 + 2*a*c*d^4*e^11 + a^2*d^2*e^1
3)*x^3 + 10*(c^2*d^7*e^8 + 2*a*c*d^5*e^10 + a^2*d^3*e^12)*x^2 + 5*(c^2*d^8*e^7 +
2*a*c*d^6*e^9 + a^2*d^4*e^11)*x)*sqrt(-c*d^2 - a*e^2)), 1/240*(240*(c^4*d^9 + 2
*a*c^3*d^7*e^2 + a^2*c^2*d^5*e^4 + (c^4*d^4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)
*x^5 + 5*(c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*c^2*d*e^8)*x^4 + 10*(c^4*d^6*e^3 +
2*a*c^3*d^4*e^5 + a^2*c^2*d^2*e^7)*x^3 + 10*(c^4*d^7*e^2 + 2*a*c^3*d^5*e^4 + a^
2*c^2*d^3*e^6)*x^2 + 5*(c^4*d^8*e + 2*a*c^3*d^6*e^3 + a^2*c^2*d^4*e^5)*x)*sqrt(c
*d^2 + a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - 2*(120*c^4*d^8*e
+ 220*a*c^3*d^6*e^3 + 89*a^2*c^2*d^4*e^5 + 58*a^3*c*d^2*e^7 + 24*a^4*e^9 + (274
*c^4*d^4*e^5 + 503*a*c^3*d^2*e^7 + 184*a^2*c^2*e^9)*x^4 + 5*(154*c^4*d^5*e^4 + 2
75*a*c^3*d^3*e^6 + 85*a^2*c^2*d*e^8)*x^3 + (940*c^4*d^6*e^3 + 1743*a*c^3*d^4*e^5
+ 621*a^2*c^2*d^2*e^7 + 88*a^3*c*e^9)*x^2 + 5*(108*c^4*d^7*e^2 + 199*a*c^3*d^5*
e^4 + 65*a^2*c^2*d^3*e^6 + 10*a^3*c*d*e^8)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a
) + 15*(8*c^5*d^10 + 20*a*c^4*d^8*e^2 + 15*a^2*c^3*d^6*e^4 + (8*c^5*d^5*e^5 + 20
*a*c^4*d^3*e^7 + 15*a^2*c^3*d*e^9)*x^5 + 5*(8*c^5*d^6*e^4 + 20*a*c^4*d^4*e^6 + 1
5*a^2*c^3*d^2*e^8)*x^4 + 10*(8*c^5*d^7*e^3 + 20*a*c^4*d^5*e^5 + 15*a^2*c^3*d^3*e
^7)*x^3 + 10*(8*c^5*d^8*e^2 + 20*a*c^4*d^6*e^4 + 15*a^2*c^3*d^4*e^6)*x^2 + 5*(8*
c^5*d^9*e + 20*a*c^4*d^7*e^3 + 15*a^2*c^3*d^5*e^5)*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)))/
((c^2*d^9*e^6 + 2*a*c*d^7*e^8 + a^2*d^5*e^10 + (c^2*d^4*e^11 + 2*a*c*d^2*e^13 +
a^2*e^15)*x^5 + 5*(c^2*d^5*e^10 + 2*a*c*d^3*e^12 + a^2*d*e^14)*x^4 + 10*(c^2*d^6
*e^9 + 2*a*c*d^4*e^11 + a^2*d^2*e^13)*x^3 + 10*(c^2*d^7*e^8 + 2*a*c*d^5*e^10 + a
^2*d^3*e^12)*x^2 + 5*(c^2*d^8*e^7 + 2*a*c*d^6*e^9 + a^2*d^4*e^11)*x)*sqrt(c*d^2
+ a*e^2)), 1/120*(120*(c^4*d^9 + 2*a*c^3*d^7*e^2 + a^2*c^2*d^5*e^4 + (c^4*d^4*e^
5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x^5 + 5*(c^4*d^5*e^4 + 2*a*c^3*d^3*e^6 + a^2*
c^2*d*e^8)*x^4 + 10*(c^4*d^6*e^3 + 2*a*c^3*d^4*e^5 + a^2*c^2*d^2*e^7)*x^3 + 10*(
c^4*d^7*e^2 + 2*a*c^3*d^5*e^4 + a^2*c^2*d^3*e^6)*x^2 + 5*(c^4*d^8*e + 2*a*c^3*d^
6*e^3 + a^2*c^2*d^4*e^5)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2
+ a)*sqrt(-c))) - (120*c^4*d^8*e + 220*a*c^3*d^6*e^3 + 89*a^2*c^2*d^4*e^5 + 58*
a^3*c*d^2*e^7 + 24*a^4*e^9 + (274*c^4*d^4*e^5 + 503*a*c^3*d^2*e^7 + 184*a^2*c^2*
e^9)*x^4 + 5*(154*c^4*d^5*e^4 + 275*a*c^3*d^3*e^6 + 85*a^2*c^2*d*e^8)*x^3 + (940
*c^4*d^6*e^3 + 1743*a*c^3*d^4*e^5 + 621*a^2*c^2*d^2*e^7 + 88*a^3*c*e^9)*x^2 + 5*
(108*c^4*d^7*e^2 + 199*a*c^3*d^5*e^4 + 65*a^2*c^2*d^3*e^6 + 10*a^3*c*d*e^8)*x)*s
qrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 15*(8*c^5*d^10 + 20*a*c^4*d^8*e^2 + 15*a^2
*c^3*d^6*e^4 + (8*c^5*d^5*e^5 + 20*a*c^4*d^3*e^7 + 15*a^2*c^3*d*e^9)*x^5 + 5*(8*
c^5*d^6*e^4 + 20*a*c^4*d^4*e^6 + 15*a^2*c^3*d^2*e^8)*x^4 + 10*(8*c^5*d^7*e^3 + 2
0*a*c^4*d^5*e^5 + 15*a^2*c^3*d^3*e^7)*x^3 + 10*(8*c^5*d^8*e^2 + 20*a*c^4*d^6*e^4
+ 15*a^2*c^3*d^4*e^6)*x^2 + 5*(8*c^5*d^9*e + 20*a*c^4*d^7*e^3 + 15*a^2*c^3*d^5*
e^5)*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))))/((c^2*d^9*e^6 + 2*a*c*d^7*e^8 + a^2*d^5*e^10 + (c^2*d^4*e^11 + 2*a*c*d^2*e
^13 + a^2*e^15)*x^5 + 5*(c^2*d^5*e^10 + 2*a*c*d^3*e^12 + a^2*d*e^14)*x^4 + 10*(c
^2*d^6*e^9 + 2*a*c*d^4*e^11 + a^2*d^2*e^13)*x^3 + 10*(c^2*d^7*e^8 + 2*a*c*d^5*e^
10 + a^2*d^3*e^12)*x^2 + 5*(c^2*d^8*e^7 + 2*a*c*d^6*e^9 + a^2*d^4*e^11)*x)*sqrt(
-c*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(5/2)/(e*x+d)**6,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**6, x)

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GIAC/XCAS [A]  time = 0.649382, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x