3.1171 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^6} \, dx\)
Optimal. Leaf size=449 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (B d \left (-15 b^2 e^2+42 b c d e+8 c^2 d^2\right )-A e \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}-\frac{b^2 \left (b^3 \left (-e^2\right ) (7 A e+3 B d)+6 b^2 c d e (5 A e+2 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}+\frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (7 A e+3 B d)+6 b^2 c d e (5 A e+2 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right )}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{\left (b x+c x^2\right )^{3/2} (7 A e (2 c d-b e)-B d (3 b e+4 c d))}{40 d^2 (d+e x)^4 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{5 d (d+e x)^5 (c d-b e)} \]
[Out]
((32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e) + 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*
e^2*(3*B*d + 7*A*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d -
b*e)^4*(d + e*x)^2) + ((B*d - A*e)*(b*x + c*x^2)^(3/2))/(5*d*(c*d - b*e)*(d + e*
x)^5) - ((7*A*e*(2*c*d - b*e) - B*d*(4*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(40*d^
2*(c*d - b*e)^2*(d + e*x)^4) + ((B*d*(8*c^2*d^2 + 42*b*c*d*e - 15*b^2*e^2) - A*e
*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2))*(b*x + c*x^2)^(3/2))/(240*d^3*(c*d -
b*e)^3*(d + e*x)^3) - (b^2*(32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e) + 6*b^2*c*
d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(
2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(256*d^(9/2)*(c*d - b*e)^(9/2))
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Rubi [A] time = 1.73849, antiderivative size = 449, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192
\[ \frac{\left (b x+c x^2\right )^{3/2} \left (B d \left (-15 b^2 e^2+42 b c d e+8 c^2 d^2\right )-A e \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}-\frac{b^2 \left (b^3 \left (-e^2\right ) (7 A e+3 B d)+6 b^2 c d e (5 A e+2 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}+\frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^3 \left (-e^2\right ) (7 A e+3 B d)+6 b^2 c d e (5 A e+2 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right )}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{\left (b x+c x^2\right )^{3/2} (7 A e (2 c d-b e)-B d (3 b e+4 c d))}{40 d^2 (d+e x)^4 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{5 d (d+e x)^5 (c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^6,x]
[Out]
((32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e) + 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*
e^2*(3*B*d + 7*A*e))*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d -
b*e)^4*(d + e*x)^2) + ((B*d - A*e)*(b*x + c*x^2)^(3/2))/(5*d*(c*d - b*e)*(d + e*
x)^5) - ((7*A*e*(2*c*d - b*e) - B*d*(4*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(40*d^
2*(c*d - b*e)^2*(d + e*x)^4) + ((B*d*(8*c^2*d^2 + 42*b*c*d*e - 15*b^2*e^2) - A*e
*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2))*(b*x + c*x^2)^(3/2))/(240*d^3*(c*d -
b*e)^3*(d + e*x)^3) - (b^2*(32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e) + 6*b^2*c*
d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(
2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(256*d^(9/2)*(c*d - b*e)^(9/2))
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**6,x)
[Out]
Timed out
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Mathematica [A] time = 4.36066, size = 535, normalized size = 1.19 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^2 \left (b^3 e^2 (7 A e+3 B d)-6 b^2 c d e (5 A e+2 B d)+16 b c^2 d^2 (3 A e+B d)-32 A c^3 d^3\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x} \sqrt{b e-c d}}+\frac{\sqrt{d} \sqrt{x} \left (8 d^2 (d+e x)^2 (c d-b e)^2 \left (A e \left (7 b^2 e^2-12 b c d e+12 c^2 d^2\right )+B d \left (3 b^2 e^2-18 b c d e+8 c^2 d^2\right )\right )+2 d (d+e x)^3 (c d-b e) \left (A e \left (-35 b^3 e^3+94 b^2 c d e^2-72 b c^2 d^2 e+48 c^3 d^3\right )+B d \left (-15 b^3 e^3+36 b^2 c d e^2-88 b c^2 d^2 e+32 c^3 d^3\right )\right )+(d+e x)^4 \left (A e \left (105 b^4 e^4-380 b^3 c d e^3+476 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+96 c^4 d^4\right )+B d \left (45 b^4 e^4-150 b^3 c d e^3+144 b^2 c^2 d^2 e^2-208 b c^3 d^3 e+64 c^4 d^4\right )\right )+384 d^4 (B d-A e) (c d-b e)^4-48 d^3 (d+e x) (c d-b e)^3 (A e (b e-2 c d)+B d (12 c d-11 b e))\right )}{e^2 (d+e x)^5}\right )}{1920 d^{9/2} \sqrt{x} (c d-b e)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^6,x]
[Out]
(Sqrt[x*(b + c*x)]*((Sqrt[d]*Sqrt[x]*(384*d^4*(B*d - A*e)*(c*d - b*e)^4 - 48*d^3
*(c*d - b*e)^3*(B*d*(12*c*d - 11*b*e) + A*e*(-2*c*d + b*e))*(d + e*x) + 8*d^2*(c
*d - b*e)^2*(B*d*(8*c^2*d^2 - 18*b*c*d*e + 3*b^2*e^2) + A*e*(12*c^2*d^2 - 12*b*c
*d*e + 7*b^2*e^2))*(d + e*x)^2 + 2*d*(c*d - b*e)*(A*e*(48*c^3*d^3 - 72*b*c^2*d^2
*e + 94*b^2*c*d*e^2 - 35*b^3*e^3) + B*d*(32*c^3*d^3 - 88*b*c^2*d^2*e + 36*b^2*c*
d*e^2 - 15*b^3*e^3))*(d + e*x)^3 + (B*d*(64*c^4*d^4 - 208*b*c^3*d^3*e + 144*b^2*
c^2*d^2*e^2 - 150*b^3*c*d*e^3 + 45*b^4*e^4) + A*e*(96*c^4*d^4 - 192*b*c^3*d^3*e
+ 476*b^2*c^2*d^2*e^2 - 380*b^3*c*d*e^3 + 105*b^4*e^4))*(d + e*x)^4))/(e^2*(d +
e*x)^5) + (15*b^2*(-32*A*c^3*d^3 + 16*b*c^2*d^2*(B*d + 3*A*e) - 6*b^2*c*d*e*(2*B
*d + 5*A*e) + b^3*e^2*(3*B*d + 7*A*e))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt
[d]*Sqrt[b + c*x])])/(Sqrt[-(c*d) + b*e]*Sqrt[b + c*x])))/(1920*d^(9/2)*(c*d - b
*e)^4*Sqrt[x])
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Maple [B] time = 0.034, size = 15015, normalized size = 33.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(1/2)/(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(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^6,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.33207, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^6,x, algorithm="fricas")
[Out]
[-1/3840*(2*(105*A*b^4*d^4*e^3 + 240*(B*b^2*c^2 - 2*A*b*c^3)*d^7 - 180*(B*b^3*c
- 4*A*b^2*c^2)*d^6*e + 45*(B*b^4 - 10*A*b^3*c)*d^5*e^2 - (64*B*c^4*d^5*e^2 + 105
*A*b^4*e^7 - 16*(13*B*b*c^3 - 6*A*c^4)*d^4*e^3 + 48*(3*B*b^2*c^2 - 4*A*b*c^3)*d^
3*e^4 - 2*(75*B*b^3*c - 238*A*b^2*c^2)*d^2*e^5 + 5*(9*B*b^4 - 76*A*b^3*c)*d*e^6)
*x^4 - 2*(160*B*c^4*d^6*e + 245*A*b^4*d*e^6 - 8*(67*B*b*c^3 - 30*A*c^4)*d^5*e^2
+ 4*(103*B*b^2*c^2 - 126*A*b*c^3)*d^4*e^3 - 13*(27*B*b^3*c - 86*A*b^2*c^2)*d^3*e
^4 + 7*(15*B*b^4 - 127*A*b^3*c)*d^2*e^5)*x^3 - 2*(320*B*c^4*d^7 + 448*A*b^4*d^2*
e^5 - 160*(7*B*b*c^3 - 3*A*c^4)*d^6*e + 8*(124*B*b^2*c^2 - 135*A*b*c^3)*d^5*e^2
- (699*B*b^3*c - 2098*A*b^2*c^2)*d^4*e^3 + (192*B*b^4 - 1631*A*b^3*c)*d^3*e^4)*x
^2 - 10*(79*A*b^4*d^3*e^4 + 16*(B*b*c^3 + 6*A*c^4)*d^7 - 4*(31*B*b^2*c^2 + 60*A*
b*c^3)*d^6*e + 3*(29*B*b^3*c + 134*A*b^2*c^2)*d^5*e^2 - (21*B*b^4 + 295*A*b^3*c)
*d^4*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 15*(7*A*b^5*d^5*e^3 + 16*(B
*b^3*c^2 - 2*A*b^2*c^3)*d^8 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^7*e + 3*(B*b^5 - 10*A
*b^4*c)*d^6*e^2 + (7*A*b^5*e^8 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3*e^5 - 12*(B*b^
4*c - 4*A*b^3*c^2)*d^2*e^6 + 3*(B*b^5 - 10*A*b^4*c)*d*e^7)*x^5 + 5*(7*A*b^5*d*e^
7 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4*e^4 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^3*e^5 +
3*(B*b^5 - 10*A*b^4*c)*d^2*e^6)*x^4 + 10*(7*A*b^5*d^2*e^6 + 16*(B*b^3*c^2 - 2*A*
b^2*c^3)*d^5*e^3 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^4*e^4 + 3*(B*b^5 - 10*A*b^4*c)*d
^3*e^5)*x^3 + 10*(7*A*b^5*d^3*e^5 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^6*e^2 - 12*(B
*b^4*c - 4*A*b^3*c^2)*d^5*e^3 + 3*(B*b^5 - 10*A*b^4*c)*d^4*e^4)*x^2 + 5*(7*A*b^5
*d^4*e^4 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^7*e - 12*(B*b^4*c - 4*A*b^3*c^2)*d^6*e
^2 + 3*(B*b^5 - 10*A*b^4*c)*d^5*e^3)*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)
+ sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((c^4*d^13 - 4*b*c^3
*d^12*e + 6*b^2*c^2*d^11*e^2 - 4*b^3*c*d^10*e^3 + b^4*d^9*e^4 + (c^4*d^8*e^5 - 4
*b*c^3*d^7*e^6 + 6*b^2*c^2*d^6*e^7 - 4*b^3*c*d^5*e^8 + b^4*d^4*e^9)*x^5 + 5*(c^4
*d^9*e^4 - 4*b*c^3*d^8*e^5 + 6*b^2*c^2*d^7*e^6 - 4*b^3*c*d^6*e^7 + b^4*d^5*e^8)*
x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 + 6*b^2*c^2*d^8*e^5 - 4*b^3*c*d^7*e^6 +
b^4*d^6*e^7)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 + 6*b^2*c^2*d^9*e^4 - 4*
b^3*c*d^8*e^5 + b^4*d^7*e^6)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*d^11*e^2 + 6*b^2*c^2*
d^10*e^3 - 4*b^3*c*d^9*e^4 + b^4*d^8*e^5)*x)*sqrt(c*d^2 - b*d*e)), -1/1920*((105
*A*b^4*d^4*e^3 + 240*(B*b^2*c^2 - 2*A*b*c^3)*d^7 - 180*(B*b^3*c - 4*A*b^2*c^2)*d
^6*e + 45*(B*b^4 - 10*A*b^3*c)*d^5*e^2 - (64*B*c^4*d^5*e^2 + 105*A*b^4*e^7 - 16*
(13*B*b*c^3 - 6*A*c^4)*d^4*e^3 + 48*(3*B*b^2*c^2 - 4*A*b*c^3)*d^3*e^4 - 2*(75*B*
b^3*c - 238*A*b^2*c^2)*d^2*e^5 + 5*(9*B*b^4 - 76*A*b^3*c)*d*e^6)*x^4 - 2*(160*B*
c^4*d^6*e + 245*A*b^4*d*e^6 - 8*(67*B*b*c^3 - 30*A*c^4)*d^5*e^2 + 4*(103*B*b^2*c
^2 - 126*A*b*c^3)*d^4*e^3 - 13*(27*B*b^3*c - 86*A*b^2*c^2)*d^3*e^4 + 7*(15*B*b^4
- 127*A*b^3*c)*d^2*e^5)*x^3 - 2*(320*B*c^4*d^7 + 448*A*b^4*d^2*e^5 - 160*(7*B*b
*c^3 - 3*A*c^4)*d^6*e + 8*(124*B*b^2*c^2 - 135*A*b*c^3)*d^5*e^2 - (699*B*b^3*c -
2098*A*b^2*c^2)*d^4*e^3 + (192*B*b^4 - 1631*A*b^3*c)*d^3*e^4)*x^2 - 10*(79*A*b^
4*d^3*e^4 + 16*(B*b*c^3 + 6*A*c^4)*d^7 - 4*(31*B*b^2*c^2 + 60*A*b*c^3)*d^6*e + 3
*(29*B*b^3*c + 134*A*b^2*c^2)*d^5*e^2 - (21*B*b^4 + 295*A*b^3*c)*d^4*e^3)*x)*sqr
t(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 15*(7*A*b^5*d^5*e^3 + 16*(B*b^3*c^2 - 2*A*
b^2*c^3)*d^8 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^7*e + 3*(B*b^5 - 10*A*b^4*c)*d^6*e^2
+ (7*A*b^5*e^8 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3*e^5 - 12*(B*b^4*c - 4*A*b^3*c
^2)*d^2*e^6 + 3*(B*b^5 - 10*A*b^4*c)*d*e^7)*x^5 + 5*(7*A*b^5*d*e^7 + 16*(B*b^3*c
^2 - 2*A*b^2*c^3)*d^4*e^4 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^3*e^5 + 3*(B*b^5 - 10*A
*b^4*c)*d^2*e^6)*x^4 + 10*(7*A*b^5*d^2*e^6 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^5*e^
3 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^4*e^4 + 3*(B*b^5 - 10*A*b^4*c)*d^3*e^5)*x^3 + 1
0*(7*A*b^5*d^3*e^5 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^6*e^2 - 12*(B*b^4*c - 4*A*b^
3*c^2)*d^5*e^3 + 3*(B*b^5 - 10*A*b^4*c)*d^4*e^4)*x^2 + 5*(7*A*b^5*d^4*e^4 + 16*(
B*b^3*c^2 - 2*A*b^2*c^3)*d^7*e - 12*(B*b^4*c - 4*A*b^3*c^2)*d^6*e^2 + 3*(B*b^5 -
10*A*b^4*c)*d^5*e^3)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d -
b*e)*x)))/((c^4*d^13 - 4*b*c^3*d^12*e + 6*b^2*c^2*d^11*e^2 - 4*b^3*c*d^10*e^3 +
b^4*d^9*e^4 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 + 6*b^2*c^2*d^6*e^7 - 4*b^3*c*d^5*e
^8 + b^4*d^4*e^9)*x^5 + 5*(c^4*d^9*e^4 - 4*b*c^3*d^8*e^5 + 6*b^2*c^2*d^7*e^6 - 4
*b^3*c*d^6*e^7 + b^4*d^5*e^8)*x^4 + 10*(c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 + 6*b^2*c
^2*d^8*e^5 - 4*b^3*c*d^7*e^6 + b^4*d^6*e^7)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^1
0*e^3 + 6*b^2*c^2*d^9*e^4 - 4*b^3*c*d^8*e^5 + b^4*d^7*e^6)*x^2 + 5*(c^4*d^12*e -
4*b*c^3*d^11*e^2 + 6*b^2*c^2*d^10*e^3 - 4*b^3*c*d^9*e^4 + b^4*d^8*e^5)*x)*sqrt(
-c*d^2 + b*d*e))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d)**6,x)
[Out]
Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**6, x)
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GIAC/XCAS [A] time = 0.611335, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d)^6,x, algorithm="giac")
[Out]
sage0*x