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

Optimal. Leaf size=356 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{48 c e^3}+\frac{\sqrt{b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{128 c^2 e^5}-\frac{(2 c d-b e) \left (3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2} e^6}+\frac{d^{5/2} (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^6}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e} \]

[Out]

((128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e
^4 - 2*c*e*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e - 3*b^2*e^2)*x)*Sqrt[b*x + c*x
^2])/(128*c^2*e^5) + ((16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2 - 6*c*e*(2*c*d - b*e)
*x)*(b*x + c*x^2)^(3/2))/(48*c*e^3) + (b*x + c*x^2)^(5/2)/(5*e) - ((2*c*d - b*e)
*(128*c^4*d^4 - 256*b*c^3*d^3*e + 112*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + 3*b^4*e
^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(5/2)*e^6) + (d^(5/2)*(c*d -
b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x +
c*x^2])])/e^6

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Rubi [A]  time = 1.09627, antiderivative size = 356, 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 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{48 c e^3}+\frac{\sqrt{b x+c x^2} \left (-3 b^4 e^4-10 b^3 c d e^3-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )}{128 c^2 e^5}-\frac{(2 c d-b e) \left (3 b^4 e^4+16 b^3 c d e^3+112 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2} e^6}+\frac{d^{5/2} (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^6}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

((128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e
^4 - 2*c*e*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e - 3*b^2*e^2)*x)*Sqrt[b*x + c*x
^2])/(128*c^2*e^5) + ((16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2 - 6*c*e*(2*c*d - b*e)
*x)*(b*x + c*x^2)^(3/2))/(48*c*e^3) + (b*x + c*x^2)^(5/2)/(5*e) - ((2*c*d - b*e)
*(128*c^4*d^4 - 256*b*c^3*d^3*e + 112*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + 3*b^4*e
^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(128*c^(5/2)*e^6) + (d^(5/2)*(c*d -
b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x +
c*x^2])])/e^6

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Rubi in Sympy [A]  time = 149.826, size = 354, normalized size = 0.99 \[ - \frac{d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{e^{6}} + \frac{\left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 e} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{3 b^{2} e^{2}}{2} - 11 b c d e + 8 c^{2} d^{2} + 3 c e x \left (b e - 2 c d\right )\right )}{24 c e^{3}} - \frac{\sqrt{b x + c x^{2}} \left (\frac{3 b^{4} e^{4}}{4} + \frac{5 b^{3} c d e^{3}}{2} - 44 b^{2} c^{2} d^{2} e^{2} + 72 b c^{3} d^{3} e - 32 c^{4} d^{4} + \frac{c e x \left (b e - 2 c d\right ) \left (3 b^{2} e^{2} + 16 b c d e - 16 c^{2} d^{2}\right )}{2}\right )}{32 c^{2} e^{5}} + \frac{\left (b e - 2 c d\right ) \left (3 b^{4} e^{4} + 16 b^{3} c d e^{3} + 112 b^{2} c^{2} d^{2} e^{2} - 256 b c^{3} d^{3} e + 128 c^{4} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{128 c^{\frac{5}{2}} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**(5/2)*(b*e - c*d)**(5/2)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e -
c*d)*sqrt(b*x + c*x**2)))/e**6 + (b*x + c*x**2)**(5/2)/(5*e) + (b*x + c*x**2)**
(3/2)*(3*b**2*e**2/2 - 11*b*c*d*e + 8*c**2*d**2 + 3*c*e*x*(b*e - 2*c*d))/(24*c*e
**3) - sqrt(b*x + c*x**2)*(3*b**4*e**4/4 + 5*b**3*c*d*e**3/2 - 44*b**2*c**2*d**2
*e**2 + 72*b*c**3*d**3*e - 32*c**4*d**4 + c*e*x*(b*e - 2*c*d)*(3*b**2*e**2 + 16*
b*c*d*e - 16*c**2*d**2)/2)/(32*c**2*e**5) + (b*e - 2*c*d)*(3*b**4*e**4 + 16*b**3
*c*d*e**3 + 112*b**2*c**2*d**2*e**2 - 256*b*c**3*d**3*e + 128*c**4*d**4)*atanh(s
qrt(c)*x/sqrt(b*x + c*x**2))/(128*c**(5/2)*e**6)

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Mathematica [A]  time = 0.954752, size = 345, normalized size = 0.97 \[ \frac{(x (b+c x))^{5/2} \left (\frac{e \sqrt{x} \left (-45 b^4 e^4+30 b^3 c e^3 (e x-5 d)+4 b^2 c^2 e^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )+16 b c^3 e \left (-270 d^3+130 d^2 e x-85 d e^2 x^2+63 e^3 x^3\right )+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )}{c^2 (b+c x)^2}-\frac{15 \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{c^{5/2} (b+c x)^{5/2}}-\frac{3840 d^{5/2} (b e-c d)^{5/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{5/2}}\right )}{1920 e^6 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(-45*b^4*e^4 + 30*b^3*c*e^3*(-5*d + e*x) + 4*b^
2*c^2*e^2*(660*d^2 - 295*d*e*x + 186*e^2*x^2) + 16*b*c^3*e*(-270*d^3 + 130*d^2*e
*x - 85*d*e^2*x^2 + 63*e^3*x^3) + 32*c^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 -
15*d*e^3*x^3 + 12*e^4*x^4)))/(c^2*(b + c*x)^2) - (3840*d^(5/2)*(-(c*d) + b*e)^(
5/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(b + c*x)^(5/
2) - (15*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e
^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(c^(5/2
)*(b + c*x)^(5/2))))/(1920*e^6*x^(5/2))

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Maple [B]  time = 0.011, size = 1932, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/32/e^2*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*d+11/8
/e^3*d^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^2+1/e^5*d^4
*(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+1/3/e^3*d^2*(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c+1/8/e*(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+1/16/e/c*(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2-11/24/e^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(3/2)*b*d-3/128/e/c^2*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)+3/256/e/c^(5/2)*b^5*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))-1/e^6*d^5*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)+1/5/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2
)+3/e^6*d^5/(-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*c^2+3/4/e^3*d^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*b*c-3/e^5*d^4/(-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-5/64/e^2/c*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)*d-1/2/e^4*d^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*c^2-9/4/e^4*d^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*
(b*e-c*d)/e^2)^(1/2)*b*c+5/16/e^3*d^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))/c^(1/2)*b^3-15/8/e^4*d
^3*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^2-1/4/e^2*(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*d-3/64/e/c*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+5/128/e^2*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^4+5/2/e^5*d^4*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+1/e^4*d^3/(-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^3-1/e^7*d^6/(-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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.65836, 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),x, algorithm="fricas")

[Out]

[1/3840*(3840*(c^4*d^4 - 2*b*c^3*d^3*e + b^2*c^2*d^2*e^2)*sqrt(c*d^2 - b*d*e)*sq
rt(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*(384*c^4*e^5*x^4 + 1920*c^4*d^4*e - 4320*b*c^3*d^3*e^2 + 2640*b^2*c^2
*d^2*e^3 - 150*b^3*c*d*e^4 - 45*b^4*e^5 - 48*(10*c^4*d*e^4 - 21*b*c^3*e^5)*x^3 +
8*(80*c^4*d^2*e^3 - 170*b*c^3*d*e^4 + 93*b^2*c^2*e^5)*x^2 - 10*(96*c^4*d^3*e^2
- 208*b*c^3*d^2*e^3 + 118*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt
(c) - 15*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e
^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c
))/(c^(5/2)*e^6), 1/3840*(7680*(c^4*d^4 - 2*b*c^3*d^3*e + b^2*c^2*d^2*e^2)*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
*(384*c^4*e^5*x^4 + 1920*c^4*d^4*e - 4320*b*c^3*d^3*e^2 + 2640*b^2*c^2*d^2*e^3 -
150*b^3*c*d*e^4 - 45*b^4*e^5 - 48*(10*c^4*d*e^4 - 21*b*c^3*e^5)*x^3 + 8*(80*c^4
*d^2*e^3 - 170*b*c^3*d*e^4 + 93*b^2*c^2*e^5)*x^2 - 10*(96*c^4*d^3*e^2 - 208*b*c^
3*d^2*e^3 + 118*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(c) - 15*(
256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 10*b^
4*c*d*e^4 - 3*b^5*e^5)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/(c^(5/2
)*e^6), 1/1920*(1920*(c^4*d^4 - 2*b*c^3*d^3*e + b^2*c^2*d^2*e^2)*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)) + (384*c^4*e^5*x^4 + 1920*c^4*d^4*e - 4320*b*c^3*d^3*e^2 + 2640*b
^2*c^2*d^2*e^3 - 150*b^3*c*d*e^4 - 45*b^4*e^5 - 48*(10*c^4*d*e^4 - 21*b*c^3*e^5)
*x^3 + 8*(80*c^4*d^2*e^3 - 170*b*c^3*d*e^4 + 93*b^2*c^2*e^5)*x^2 - 10*(96*c^4*d^
3*e^2 - 208*b*c^3*d^2*e^3 + 118*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x
)*sqrt(-c) - 15*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^
2*d^2*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x))
)/(sqrt(-c)*c^2*e^6), 1/1920*(3840*(c^4*d^4 - 2*b*c^3*d^3*e + b^2*c^2*d^2*e^2)*s
qrt(-c*d^2 + b*d*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)
) + (384*c^4*e^5*x^4 + 1920*c^4*d^4*e - 4320*b*c^3*d^3*e^2 + 2640*b^2*c^2*d^2*e^
3 - 150*b^3*c*d*e^4 - 45*b^4*e^5 - 48*(10*c^4*d*e^4 - 21*b*c^3*e^5)*x^3 + 8*(80*
c^4*d^2*e^3 - 170*b*c^3*d*e^4 + 93*b^2*c^2*e^5)*x^2 - 10*(96*c^4*d^3*e^2 - 208*b
*c^3*d^2*e^3 + 118*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) -
15*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 1
0*b^4*c*d*e^4 - 3*b^5*e^5)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c
^2*e^6)]

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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),x)

[Out]

Timed out

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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 + b*x)^(5/2)/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError