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

Optimal. Leaf size=282 \[ -\frac{5 (2 c d-b e) \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 \sqrt{c} e^6}+\frac{5 \sqrt{b x+c x^2} \left (5 b^2 e^2-4 c e x (2 c d-b e)-20 b c d e+16 c^2 d^2\right )}{8 e^5}+\frac{5 \sqrt{d} (4 c d-3 b e) \sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 e^6}+\frac{5 \left (b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{12 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{5/2}}{2 e (d+e x)^2} \]

[Out]

(5*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2 - 4*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^
2])/(8*e^5) + (5*(8*c*d - 3*b*e + 2*c*e*x)*(b*x + c*x^2)^(3/2))/(12*e^3*(d + e*x
)) - (b*x + c*x^2)^(5/2)/(2*e*(d + e*x)^2) - (5*(2*c*d - b*e)*(16*c^2*d^2 - 16*b
*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*Sqrt[c]*e^6) + (5*S
qrt[d]*(4*c*d - 3*b*e)*Sqrt[c*d - b*e]*(4*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])])/(8*e^6)

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Rubi [A]  time = 0.871426, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5 (2 c d-b e) \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 \sqrt{c} e^6}+\frac{5 \sqrt{b x+c x^2} \left (5 b^2 e^2-4 c e x (2 c d-b e)-20 b c d e+16 c^2 d^2\right )}{8 e^5}+\frac{5 \sqrt{d} (4 c d-3 b e) \sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 e^6}+\frac{5 \left (b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{12 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{5/2}}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(5*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2 - 4*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^
2])/(8*e^5) + (5*(8*c*d - 3*b*e + 2*c*e*x)*(b*x + c*x^2)^(3/2))/(12*e^3*(d + e*x
)) - (b*x + c*x^2)^(5/2)/(2*e*(d + e*x)^2) - (5*(2*c*d - b*e)*(16*c^2*d^2 - 16*b
*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*Sqrt[c]*e^6) + (5*S
qrt[d]*(4*c*d - 3*b*e)*Sqrt[c*d - b*e]*(4*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])])/(8*e^6)

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Rubi in Sympy [A]  time = 111.728, size = 270, normalized size = 0.96 \[ - \frac{5 \sqrt{d} \left (b e - 4 c d\right ) \sqrt{b e - c d} \left (3 b e - 4 c d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{8 e^{6}} - \frac{\left (b x + c x^{2}\right )^{\frac{5}{2}}}{2 e \left (d + e x\right )^{2}} - \frac{5 \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (3 b e - 8 c d - 2 c e x\right )}{12 e^{3} \left (d + e x\right )} + \frac{5 \sqrt{b x + c x^{2}} \left (10 b^{2} e^{2} - 40 b c d e + 32 c^{2} d^{2} + 8 c e x \left (b e - 2 c d\right )\right )}{16 e^{5}} + \frac{5 \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 \sqrt{c} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(d)*(b*e - 4*c*d)*sqrt(b*e - c*d)*(3*b*e - 4*c*d)*atan((-b*d + x*(b*e - 2
*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(8*e**6) - (b*x + c*x**2)
**(5/2)/(2*e*(d + e*x)**2) - 5*(b*x + c*x**2)**(3/2)*(3*b*e - 8*c*d - 2*c*e*x)/(
12*e**3*(d + e*x)) + 5*sqrt(b*x + c*x**2)*(10*b**2*e**2 - 40*b*c*d*e + 32*c**2*d
**2 + 8*c*e*x*(b*e - 2*c*d))/(16*e**5) + 5*(b*e - 2*c*d)*(b**2*e**2 - 16*b*c*d*e
+ 16*c**2*d**2)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*sqrt(c)*e**6)

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Mathematica [A]  time = 1.20419, size = 329, normalized size = 1.17 \[ \frac{(x (b+c x))^{5/2} \left (\frac{e \sqrt{x} \left (3 b^2 e^2 \left (25 d^2+40 d e x+11 e^2 x^2\right )-2 b c e \left (150 d^3+230 d^2 e x+55 d e^2 x^2-13 e^3 x^3\right )+4 c^2 \left (60 d^4+90 d^3 e x+20 d^2 e^2 x^2-5 d e^3 x^3+2 e^4 x^4\right )\right )}{(b+c x)^2 (d+e x)^2}-\frac{15 \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} (b+c x)^{5/2}}+\frac{30 \sqrt{d} \left (-3 b^3 e^3+19 b^2 c d e^2-32 b c^2 d^2 e+16 c^3 d^3\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{5/2} \sqrt{b e-c d}}\right )}{24 e^6 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(5/2)*((e*Sqrt[x]*(3*b^2*e^2*(25*d^2 + 40*d*e*x + 11*e^2*x^2) - 2
*b*c*e*(150*d^3 + 230*d^2*e*x + 55*d*e^2*x^2 - 13*e^3*x^3) + 4*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)))/((b + c*x)^2*(d + e*x)^2)
+ (30*Sqrt[d]*(16*c^3*d^3 - 32*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 3*b^3*e^3)*ArcTan[
(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[-(c*d) + b*e]*(b +
c*x)^(5/2)) - (15*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*e^3)*Log[c
*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[c]*(b + c*x)^(5/2))))/(24*e^6*x^(5/2))

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Maple [B]  time = 0.018, size = 5534, normalized size = 19.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/2)/(e*x+d)^3,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 + b*x)^(5/2)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.501865, 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)^3,x, algorithm="fricas")

[Out]

[1/48*(30*(16*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (16*c^2*d^2*e^2 - 16*b*c*
d*e^3 + 3*b^2*e^4)*x^2 + 2*(16*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*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*(8*c^2*e^5*x^4 + 240*c^2*d^4*e - 300*b*c*d^3*e^2 +
75*b^2*d^2*e^3 - 2*(10*c^2*d*e^4 - 13*b*c*e^5)*x^3 + (80*c^2*d^2*e^3 - 110*b*c*d
*e^4 + 33*b^2*e^5)*x^2 + 20*(18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 6*b^2*d*e^4)*x)*s
qrt(c*x^2 + b*x)*sqrt(c) - 15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 18*b^2*c*d^3*e^2 -
b^3*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 18*b^2*c*d*e^4 - b^3*e^5)*x^2
+ 2*(32*c^3*d^4*e - 48*b*c^2*d^3*e^2 + 18*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*log((2*
c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt
(c)), 1/48*(60*(16*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (16*c^2*d^2*e^2 - 16
*b*c*d*e^3 + 3*b^2*e^4)*x^2 + 2*(16*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*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*(8*c^2*e^5*x^4 + 240*c^2*d^4*e - 300*b*c*d^3*e^2 + 75*b^2*d^2*e^3 - 2*(10
*c^2*d*e^4 - 13*b*c*e^5)*x^3 + (80*c^2*d^2*e^3 - 110*b*c*d*e^4 + 33*b^2*e^5)*x^2
+ 20*(18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 6*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(
c) - 15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 18*b^2*c*d^3*e^2 - b^3*d^2*e^3 + (32*c^3*
d^3*e^2 - 48*b*c^2*d^2*e^3 + 18*b^2*c*d*e^4 - b^3*e^5)*x^2 + 2*(32*c^3*d^4*e - 4
8*b*c^2*d^3*e^2 + 18*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*log((2*c*x + b)*sqrt(c) + 2*s
qrt(c*x^2 + b*x)*c))/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(c)), 1/24*(15*(16*c^2
*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (16*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4
)*x^2 + 2*(16*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*s
qrt(-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)) + (8*c^2*e^5*x^4 + 240*c^2*d^4*e - 300*b*c*d^3*e^2 + 75*b^2*d^2*e^3 - 2
*(10*c^2*d*e^4 - 13*b*c*e^5)*x^3 + (80*c^2*d^2*e^3 - 110*b*c*d*e^4 + 33*b^2*e^5)
*x^2 + 20*(18*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 6*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x)*s
qrt(-c) - 15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 18*b^2*c*d^3*e^2 - b^3*d^2*e^3 + (32
*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 18*b^2*c*d*e^4 - b^3*e^5)*x^2 + 2*(32*c^3*d^4*
e - 48*b*c^2*d^3*e^2 + 18*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*arctan(sqrt(c*x^2 + b*x)
*sqrt(-c)/(c*x)))/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(-c)), 1/24*(30*(16*c^2*d
^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (16*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*
x^2 + 2*(16*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sq
rt(-c)*arctan(sqrt(c*x^2 + b*x)*d/(sqrt(-c*d^2 + b*d*e)*x)) + (8*c^2*e^5*x^4 + 2
40*c^2*d^4*e - 300*b*c*d^3*e^2 + 75*b^2*d^2*e^3 - 2*(10*c^2*d*e^4 - 13*b*c*e^5)*
x^3 + (80*c^2*d^2*e^3 - 110*b*c*d*e^4 + 33*b^2*e^5)*x^2 + 20*(18*c^2*d^3*e^2 - 2
3*b*c*d^2*e^3 + 6*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 15*(32*c^3*d^5 - 48
*b*c^2*d^4*e + 18*b^2*c*d^3*e^2 - b^3*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c^2*d^2*e
^3 + 18*b^2*c*d*e^4 - b^3*e^5)*x^2 + 2*(32*c^3*d^4*e - 48*b*c^2*d^3*e^2 + 18*b^2
*c*d^2*e^3 - b^3*d*e^4)*x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/((e^8*x^2 +
2*d*e^7*x + d^2*e^6)*sqrt(-c))]

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.626333, 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 + b*x)^(5/2)/(e*x + d)^3,x, algorithm="giac")

[Out]

sage0*x