3.307 \(\int \frac{\left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx\)
Optimal. Leaf size=314 \[ -\frac{5 \sqrt{b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{64 c e^5}+\frac{5 \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac{5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^6}-\frac{5 \left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)} \]
[Out]
(-5*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3 - 2*c*e*(16*c^2*d^2
- 16*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e -
6*c*e*x)*(b*x + c*x^2)^(3/2))/(24*e^3) - (b*x + c*x^2)^(5/2)/(e*(d + e*x)) + (5*
(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4)
*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^6) - (5*d^(3/2)*(c*d - b*
e)^(3/2)*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e
]*Sqrt[b*x + c*x^2])])/(2*e^6)
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Rubi [A] time = 1.01058, antiderivative size = 314, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286
\[ -\frac{5 \sqrt{b x+c x^2} \left (-b^3 e^3-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )}{64 c e^5}+\frac{5 \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac{5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^6}-\frac{5 \left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(5/2)/(d + e*x)^2,x]
[Out]
(-5*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3 - 2*c*e*(16*c^2*d^2
- 16*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e -
6*c*e*x)*(b*x + c*x^2)^(3/2))/(24*e^3) - (b*x + c*x^2)^(5/2)/(e*(d + e*x)) + (5*
(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4)
*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^6) - (5*d^(3/2)*(c*d - b*
e)^(3/2)*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e
]*Sqrt[b*x + c*x^2])])/(2*e^6)
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Rubi in Sympy [A] time = 139.82, size = 304, normalized size = 0.97 \[ \frac{5 d^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{2 e^{6}} - \frac{\left (b x + c x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )} + \frac{5 \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (7 b e - 8 c d + 6 c e x\right )}{24 e^{3}} + \frac{5 \sqrt{b x + c x^{2}} \left (\frac{b^{3} e^{3}}{2} - 24 b^{2} c d e^{2} + 56 b c^{2} d^{2} e - 32 c^{3} d^{3} + c e x \left (b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )\right )}{32 c e^{5}} - \frac{5 \left (b^{4} e^{4} + 16 b^{3} c d e^{3} - 144 b^{2} c^{2} d^{2} e^{2} + 256 b c^{3} d^{3} e - 128 c^{4} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{3}{2}} e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(5/2)/(e*x+d)**2,x)
[Out]
5*d**(3/2)*(b*e - 2*c*d)*(b*e - c*d)**(3/2)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqr
t(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(2*e**6) - (b*x + c*x**2)**(5/2)/(e*(d
+ e*x)) + 5*(b*x + c*x**2)**(3/2)*(7*b*e - 8*c*d + 6*c*e*x)/(24*e**3) + 5*sqrt(
b*x + c*x**2)*(b**3*e**3/2 - 24*b**2*c*d*e**2 + 56*b*c**2*d**2*e - 32*c**3*d**3
+ c*e*x*(b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2))/(32*c*e**5) - 5*(b**4*e**4 + 16
*b**3*c*d*e**3 - 144*b**2*c**2*d**2*e**2 + 256*b*c**3*d**3*e - 128*c**4*d**4)*at
anh(sqrt(c)*x/sqrt(b*x + c*x**2))/(64*c**(3/2)*e**6)
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Mathematica [A] time = 1.10852, size = 334, normalized size = 1.06 \[ \frac{(x (b+c x))^{5/2} \left (\frac{e \sqrt{x} \left (15 b^3 e^3 (d+e x)+2 b^2 c e^2 \left (-360 d^2-205 d e x+59 e^2 x^2\right )+8 b c^2 e \left (210 d^3+110 d^2 e x-35 d e^2 x^2+17 e^3 x^3\right )-16 c^3 \left (60 d^4+30 d^3 e x-10 d^2 e^2 x^2+5 d e^3 x^3-3 e^4 x^4\right )\right )}{c (b+c x)^2 (d+e x)}+\frac{15 \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{c^{3/2} (b+c x)^{5/2}}-\frac{960 d^{3/2} (2 c d-b e) (b e-c d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{5/2}}\right )}{192 e^6 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^2,x]
[Out]
((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(15*b^3*e^3*(d + e*x) + 2*b^2*c*e^2*(-360*d^2 -
205*d*e*x + 59*e^2*x^2) + 8*b*c^2*e*(210*d^3 + 110*d^2*e*x - 35*d*e^2*x^2 + 17*
e^3*x^3) - 16*c^3*(60*d^4 + 30*d^3*e*x - 10*d^2*e^2*x^2 + 5*d*e^3*x^3 - 3*e^4*x^
4)))/(c*(b + c*x)^2*(d + e*x)) - (960*d^(3/2)*(2*c*d - b*e)*(-(c*d) + b*e)^(3/2)
*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(b + c*x)^(5/2) +
(15*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4
*e^4)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(3/2)*(b + c*x)^(5/2))))/(192*e
^6*x^(5/2))
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Maple [B] time = 0.016, size = 2534, normalized size = 8.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(5/2)/(e*x+d)^2,x)
[Out]
-5/e^7*d^6/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)
/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-
c*d)/e^2)^(1/2))/(d/e+x))*c^4-55/4/e^4*d^3/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*
(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*b+5/e^3*d^2/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*b+25/2/e^4*d^3/(b*e-c*d)/(-d*(b*e-c*d)/
e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/
2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3*c+35/
2/e^6*d^5/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/
e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c
*d)/e^2)^(1/2))/(d/e+x))*c^3*b-85/32/e^2*d/(b*e-c*d)*b^2*(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c-45/2/e^5*d^4/(b*e-c*d)/(-d*(b*e-c*d)/e^2)
^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(
c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c^2+5/32/
e/(b*e-c*d)*b^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x+5/64
/e/(b*e-c*d)/c*b^4*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-5/1
28/e/(b*e-c*d)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*
e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^5+5/e^5*d^4/(b*e-c*d)*(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^3+5/3/e^3*d^2/(b*e-c*d)*(c*(d/e+x
)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2-245/64/e^2*d/(b*e-c*d)*(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^3-1/d/(b*e-c*d)*(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*b+1/d/(b*e-c*d)/(d/e+x)*(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(7/2)+35/24/e/(b*e-c*d)*(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2+1/e/(b*e-c*d)*(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*c-5/e^6*d^5/(b*e-c*d)*ln((1/2*(b*e-2
*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
1/2))*c^(7/2)-25/8/e^2*d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(3/2)*c*b+25/2/e^3*d^2/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(1/2)*b^2*c+5/4/e/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(3/2)*x*b*c+15/e^5*d^4/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^
(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)*b-5/4/e
^2*d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*c^2-5
/2/e^4*d^3/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x
*c^3-75/128/e^2*d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^4+25/4/e^3*d^2/(b*e-c*
d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^3-125/8/e^4*d^3/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e
+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c
^(3/2)*b^2-5/2/e^3*d^2/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2
+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^4-c/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)
/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*x
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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 + b*x)^(5/2)/(e*x + d)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.47771, 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 + b*x)^(5/2)/(e*x + d)^2,x, algorithm="fricas")
[Out]
[1/384*(960*(2*c^3*d^4 - 3*b*c^2*d^3*e + b^2*c*d^2*e^2 + (2*c^3*d^3*e - 3*b*c^2*
d^2*e^2 + b^2*c*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c)*log((b*d + (2*c*d - b*e)*x
- 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(48*c^3*e^5*x^4 - 960
*c^3*d^4*e + 1680*b*c^2*d^3*e^2 - 720*b^2*c*d^2*e^3 + 15*b^3*d*e^4 - 8*(10*c^3*d
*e^4 - 17*b*c^2*e^5)*x^3 + 2*(80*c^3*d^2*e^3 - 140*b*c^2*d*e^4 + 59*b^2*c*e^5)*x
^2 - 5*(96*c^3*d^3*e^2 - 176*b*c^2*d^2*e^3 + 82*b^2*c*d*e^4 - 3*b^3*e^5)*x)*sqrt
(c*x^2 + b*x)*sqrt(c) - 15*(128*c^4*d^5 - 256*b*c^3*d^4*e + 144*b^2*c^2*d^3*e^2
- 16*b^3*c*d^2*e^3 - b^4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^
2*d^2*e^3 - 16*b^3*c*d*e^4 - b^4*e^5)*x)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2
+ b*x)*c))/((c*e^7*x + c*d*e^6)*sqrt(c)), -1/384*(1920*(2*c^3*d^4 - 3*b*c^2*d^3*
e + b^2*c*d^2*e^2 + (2*c^3*d^3*e - 3*b*c^2*d^2*e^2 + b^2*c*d*e^3)*x)*sqrt(-c*d^2
+ b*d*e)*sqrt(c)*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - 2*(48*c
^3*e^5*x^4 - 960*c^3*d^4*e + 1680*b*c^2*d^3*e^2 - 720*b^2*c*d^2*e^3 + 15*b^3*d*e
^4 - 8*(10*c^3*d*e^4 - 17*b*c^2*e^5)*x^3 + 2*(80*c^3*d^2*e^3 - 140*b*c^2*d*e^4 +
59*b^2*c*e^5)*x^2 - 5*(96*c^3*d^3*e^2 - 176*b*c^2*d^2*e^3 + 82*b^2*c*d*e^4 - 3*
b^3*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 15*(128*c^4*d^5 - 256*b*c^3*d^4*e + 144*
b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*e^3 - b^4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*
e^2 + 144*b^2*c^2*d^2*e^3 - 16*b^3*c*d*e^4 - b^4*e^5)*x)*log((2*c*x + b)*sqrt(c)
- 2*sqrt(c*x^2 + b*x)*c))/((c*e^7*x + c*d*e^6)*sqrt(c)), 1/192*(480*(2*c^3*d^4
- 3*b*c^2*d^3*e + b^2*c*d^2*e^2 + (2*c^3*d^3*e - 3*b*c^2*d^2*e^2 + b^2*c*d*e^3)*
x)*sqrt(c*d^2 - b*d*e)*sqrt(-c)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*
e)*sqrt(c*x^2 + b*x))/(e*x + d)) + (48*c^3*e^5*x^4 - 960*c^3*d^4*e + 1680*b*c^2*
d^3*e^2 - 720*b^2*c*d^2*e^3 + 15*b^3*d*e^4 - 8*(10*c^3*d*e^4 - 17*b*c^2*e^5)*x^3
+ 2*(80*c^3*d^2*e^3 - 140*b*c^2*d*e^4 + 59*b^2*c*e^5)*x^2 - 5*(96*c^3*d^3*e^2 -
176*b*c^2*d^2*e^3 + 82*b^2*c*d*e^4 - 3*b^3*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) +
15*(128*c^4*d^5 - 256*b*c^3*d^4*e + 144*b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*e^3 - b^
4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d^2*e^3 - 16*b^3*c*d*
e^4 - b^4*e^5)*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((c*e^7*x + c*d*e^6)
*sqrt(-c)), -1/192*(960*(2*c^3*d^4 - 3*b*c^2*d^3*e + b^2*c*d^2*e^2 + (2*c^3*d^3*
e - 3*b*c^2*d^2*e^2 + b^2*c*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(-c)*arctan(sqrt(
c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) - (48*c^3*e^5*x^4 - 960*c^3*d^4*e + 168
0*b*c^2*d^3*e^2 - 720*b^2*c*d^2*e^3 + 15*b^3*d*e^4 - 8*(10*c^3*d*e^4 - 17*b*c^2*
e^5)*x^3 + 2*(80*c^3*d^2*e^3 - 140*b*c^2*d*e^4 + 59*b^2*c*e^5)*x^2 - 5*(96*c^3*d
^3*e^2 - 176*b*c^2*d^2*e^3 + 82*b^2*c*d*e^4 - 3*b^3*e^5)*x)*sqrt(c*x^2 + b*x)*sq
rt(-c) - 15*(128*c^4*d^5 - 256*b*c^3*d^4*e + 144*b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*
e^3 - b^4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d^2*e^3 - 16*
b^3*c*d*e^4 - b^4*e^5)*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((c*e^7*x +
c*d*e^6)*sqrt(-c))]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(5/2)/(e*x+d)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^2,x, algorithm="giac")
[Out]
Timed out