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

Optimal. Leaf size=287 \[ -\frac{5 c^2 \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 e^6 \left (a e^2+c d^2\right )^{3/2}}-\frac{5 c^{5/2} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^6}+\frac{5 c^2 \sqrt{a+c x^2} \left (e x \left (3 a e^2+4 c d^2\right )+8 d \left (a e^2+c d^2\right )\right )}{8 e^5 (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c \left (a+c x^2\right )^{3/2} \left (3 e x \left (a e^2+2 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{24 e^3 (d+e x)^3 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4} \]

[Out]

(5*c^2*(8*d*(c*d^2 + a*e^2) + e*(4*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5*(
c*d^2 + a*e^2)*(d + e*x)) - (5*c*(d*(4*c*d^2 + a*e^2) + 3*e*(2*c*d^2 + a*e^2)*x)
*(a + c*x^2)^(3/2))/(24*e^3*(c*d^2 + a*e^2)*(d + e*x)^3) - (a + c*x^2)^(5/2)/(4*
e*(d + e*x)^4) - (5*c^(5/2)*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^6 - (5*c^2
*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*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)^(3/2))

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Rubi [A]  time = 0.811623, antiderivative size = 287, 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 \[ -\frac{5 c^2 \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 e^6 \left (a e^2+c d^2\right )^{3/2}}-\frac{5 c^{5/2} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^6}+\frac{5 c^2 \sqrt{a+c x^2} \left (e x \left (3 a e^2+4 c d^2\right )+8 d \left (a e^2+c d^2\right )\right )}{8 e^5 (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c \left (a+c x^2\right )^{3/2} \left (3 e x \left (a e^2+2 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{24 e^3 (d+e x)^3 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{5/2}}{4 e (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(5*c^2*(8*d*(c*d^2 + a*e^2) + e*(4*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5*(
c*d^2 + a*e^2)*(d + e*x)) - (5*c*(d*(4*c*d^2 + a*e^2) + 3*e*(2*c*d^2 + a*e^2)*x)
*(a + c*x^2)^(3/2))/(24*e^3*(c*d^2 + a*e^2)*(d + e*x)^3) - (a + c*x^2)^(5/2)/(4*
e*(d + e*x)^4) - (5*c^(5/2)*d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/e^6 - (5*c^2
*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*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)^(3/2))

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Rubi in Sympy [A]  time = 86.4824, size = 270, normalized size = 0.94 \[ - \frac{5 c^{\frac{5}{2}} d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{6}} + \frac{5 c^{2} \sqrt{a + c x^{2}} \left (16 d \left (a e^{2} + c d^{2}\right ) + 2 e x \left (3 a e^{2} + 4 c d^{2}\right )\right )}{16 e^{5} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} - \frac{5 c^{2} \left (3 a^{2} e^{4} + 12 a c d^{2} e^{2} + 8 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{8 e^{6} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{5 c \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d \left (a e^{2} + 4 c d^{2}\right ) + 3 e x \left (a e^{2} + 2 c d^{2}\right )\right )}{24 e^{3} \left (d + e x\right )^{3} \left (a e^{2} + c d^{2}\right )} - \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{4 e \left (d + e x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*c**(5/2)*d*atanh(sqrt(c)*x/sqrt(a + c*x**2))/e**6 + 5*c**2*sqrt(a + c*x**2)*(
16*d*(a*e**2 + c*d**2) + 2*e*x*(3*a*e**2 + 4*c*d**2))/(16*e**5*(d + e*x)*(a*e**2
+ c*d**2)) - 5*c**2*(3*a**2*e**4 + 12*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)**(3/
2)) - 5*c*(a + c*x**2)**(3/2)*(d*(a*e**2 + 4*c*d**2) + 3*e*x*(a*e**2 + 2*c*d**2)
)/(24*e**3*(d + e*x)**3*(a*e**2 + c*d**2)) - (a + c*x**2)**(5/2)/(4*e*(d + e*x)*
*4)

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Mathematica [A]  time = 1.2201, size = 299, normalized size = 1.04 \[ \frac{-\frac{15 c^2 \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{15 c^2 \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}-120 c^{5/2} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+e \sqrt{a+c x^2} \left (\frac{c^2 d \left (139 a e^2+154 c d^2\right )}{(d+e x) \left (a e^2+c d^2\right )}-\frac{c \left (27 a e^2+86 c d^2\right )}{(d+e x)^2}+\frac{34 c d \left (a e^2+c d^2\right )}{(d+e x)^3}-\frac{6 \left (a e^2+c d^2\right )^2}{(d+e x)^4}+24 c^2\right )}{24 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[a + c*x^2]*(24*c^2 - (6*(c*d^2 + a*e^2)^2)/(d + e*x)^4 + (34*c*d*(c*d^2
+ a*e^2))/(d + e*x)^3 - (c*(86*c*d^2 + 27*a*e^2))/(d + e*x)^2 + (c^2*d*(154*c*d^
2 + 139*a*e^2))/((c*d^2 + a*e^2)*(d + e*x))) + (15*c^2*(8*c^2*d^4 + 12*a*c*d^2*e
^2 + 3*a^2*e^4)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) - 120*c^(5/2)*d*Log[c*x + Sq
rt[c]*Sqrt[a + c*x^2]] - (15*c^2*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*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)^(3/2))/(24*e^6
)

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Maple [B]  time = 0.028, size = 5406, normalized size = 18.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.2109, 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)^5,x, algorithm="fricas")

[Out]

[1/48*(120*(c^3*d^7 + a*c^2*d^5*e^2 + (c^3*d^3*e^4 + a*c^2*d*e^6)*x^4 + 4*(c^3*d
^4*e^3 + a*c^2*d^2*e^5)*x^3 + 6*(c^3*d^5*e^2 + a*c^2*d^3*e^4)*x^2 + 4*(c^3*d^6*e
+ a*c^2*d^4*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*(120*c^3*d^6*e + 100*a*c^2*d^4*e^3 - 11*a^2*c*d^2*e^5 - 6*a
^3*e^7 + 24*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 5*(50*c^3*d^3*e^4 + 47*a*c^2*d*e^6)*
x^3 + (520*c^3*d^4*e^3 + 448*a*c^2*d^2*e^5 - 27*a^2*c*e^7)*x^2 + 5*(84*c^3*d^5*e
^2 + 71*a*c^2*d^3*e^4 - 4*a^2*c*d*e^6)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) +
15*(8*c^4*d^8 + 12*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 + 12*a*c^3
*d^2*e^6 + 3*a^2*c^2*e^8)*x^4 + 4*(8*c^4*d^5*e^3 + 12*a*c^3*d^3*e^5 + 3*a^2*c^2*
d*e^7)*x^3 + 6*(8*c^4*d^6*e^2 + 12*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6)*x^2 + 4*(8
*c^4*d^7*e + 12*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*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*d^6*e^6 + a*d^4*e^8 + (c*d^2*e^10 + a*e^12)*x^4 + 4*(c*d^3*e^9 + a*d*e^11)*x
^3 + 6*(c*d^4*e^8 + a*d^2*e^10)*x^2 + 4*(c*d^5*e^7 + a*d^3*e^9)*x)*sqrt(c*d^2 +
a*e^2)), -1/48*(240*(c^3*d^7 + a*c^2*d^5*e^2 + (c^3*d^3*e^4 + a*c^2*d*e^6)*x^4 +
4*(c^3*d^4*e^3 + a*c^2*d^2*e^5)*x^3 + 6*(c^3*d^5*e^2 + a*c^2*d^3*e^4)*x^2 + 4*(
c^3*d^6*e + a*c^2*d^4*e^3)*x)*sqrt(c*d^2 + a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^
2 + a)*sqrt(-c))) - 2*(120*c^3*d^6*e + 100*a*c^2*d^4*e^3 - 11*a^2*c*d^2*e^5 - 6*
a^3*e^7 + 24*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 5*(50*c^3*d^3*e^4 + 47*a*c^2*d*e^6)
*x^3 + (520*c^3*d^4*e^3 + 448*a*c^2*d^2*e^5 - 27*a^2*c*e^7)*x^2 + 5*(84*c^3*d^5*
e^2 + 71*a*c^2*d^3*e^4 - 4*a^2*c*d*e^6)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) -
15*(8*c^4*d^8 + 12*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 + 12*a*c^
3*d^2*e^6 + 3*a^2*c^2*e^8)*x^4 + 4*(8*c^4*d^5*e^3 + 12*a*c^3*d^3*e^5 + 3*a^2*c^2
*d*e^7)*x^3 + 6*(8*c^4*d^6*e^2 + 12*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6)*x^2 + 4*(
8*c^4*d^7*e + 12*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*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*d^6*e^6 + a*d^4*e^8 + (c*d^2*e^10 + a*e^12)*x^4 + 4*(c*d^3*e^9 + a*d*e^11)*
x^3 + 6*(c*d^4*e^8 + a*d^2*e^10)*x^2 + 4*(c*d^5*e^7 + a*d^3*e^9)*x)*sqrt(c*d^2 +
a*e^2)), 1/24*(60*(c^3*d^7 + a*c^2*d^5*e^2 + (c^3*d^3*e^4 + a*c^2*d*e^6)*x^4 +
4*(c^3*d^4*e^3 + a*c^2*d^2*e^5)*x^3 + 6*(c^3*d^5*e^2 + a*c^2*d^3*e^4)*x^2 + 4*(c
^3*d^6*e + a*c^2*d^4*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) + (120*c^3*d^6*e + 100*a*c^2*d^4*e^3 - 11*a^2*c*d^2*e^
5 - 6*a^3*e^7 + 24*(c^3*d^2*e^5 + a*c^2*e^7)*x^4 + 5*(50*c^3*d^3*e^4 + 47*a*c^2*
d*e^6)*x^3 + (520*c^3*d^4*e^3 + 448*a*c^2*d^2*e^5 - 27*a^2*c*e^7)*x^2 + 5*(84*c^
3*d^5*e^2 + 71*a*c^2*d^3*e^4 - 4*a^2*c*d*e^6)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2
+ a) + 15*(8*c^4*d^8 + 12*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 +
12*a*c^3*d^2*e^6 + 3*a^2*c^2*e^8)*x^4 + 4*(8*c^4*d^5*e^3 + 12*a*c^3*d^3*e^5 + 3*
a^2*c^2*d*e^7)*x^3 + 6*(8*c^4*d^6*e^2 + 12*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6)*x^
2 + 4*(8*c^4*d^7*e + 12*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*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*d^6*e^6 + a*d^4*
e^8 + (c*d^2*e^10 + a*e^12)*x^4 + 4*(c*d^3*e^9 + a*d*e^11)*x^3 + 6*(c*d^4*e^8 +
a*d^2*e^10)*x^2 + 4*(c*d^5*e^7 + a*d^3*e^9)*x)*sqrt(-c*d^2 - a*e^2)), -1/24*(120
*(c^3*d^7 + a*c^2*d^5*e^2 + (c^3*d^3*e^4 + a*c^2*d*e^6)*x^4 + 4*(c^3*d^4*e^3 + a
*c^2*d^2*e^5)*x^3 + 6*(c^3*d^5*e^2 + a*c^2*d^3*e^4)*x^2 + 4*(c^3*d^6*e + a*c^2*d
^4*e^3)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c)))
- (120*c^3*d^6*e + 100*a*c^2*d^4*e^3 - 11*a^2*c*d^2*e^5 - 6*a^3*e^7 + 24*(c^3*d^
2*e^5 + a*c^2*e^7)*x^4 + 5*(50*c^3*d^3*e^4 + 47*a*c^2*d*e^6)*x^3 + (520*c^3*d^4*
e^3 + 448*a*c^2*d^2*e^5 - 27*a^2*c*e^7)*x^2 + 5*(84*c^3*d^5*e^2 + 71*a*c^2*d^3*e
^4 - 4*a^2*c*d*e^6)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 15*(8*c^4*d^8 + 12
*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + (8*c^4*d^4*e^4 + 12*a*c^3*d^2*e^6 + 3*a^2*c
^2*e^8)*x^4 + 4*(8*c^4*d^5*e^3 + 12*a*c^3*d^3*e^5 + 3*a^2*c^2*d*e^7)*x^3 + 6*(8*
c^4*d^6*e^2 + 12*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6)*x^2 + 4*(8*c^4*d^7*e + 12*a*
c^3*d^5*e^3 + 3*a^2*c^2*d^3*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*d^6*e^6 + a*d^4*e^8 + (c*d^2*e^10 + a*e^12
)*x^4 + 4*(c*d^3*e^9 + a*d*e^11)*x^3 + 6*(c*d^4*e^8 + a*d^2*e^10)*x^2 + 4*(c*d^5
*e^7 + a*d^3*e^9)*x)*sqrt(-c*d^2 - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError