3.1181 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx\)
Optimal. Leaf size=402 \[ -\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{512 d^4 (d+e x)^2 (c d-b e)^4}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{192 d^3 (d+e x)^4 (c d-b e)^3}+\frac{b^4 \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{60 d^2 (d+e x)^5 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)} \]
[Out]
-(b^2*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d + (2*
c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(512*d^4*(c*d - b*e)^4*(d + e*x)^2) + ((24*A*c^
2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d + (2*c*d - b*e)*x)*
(b*x + c*x^2)^(3/2))/(192*d^3*(c*d - b*e)^3*(d + e*x)^4) + ((B*d - A*e)*(b*x + c
*x^2)^(5/2))/(6*d*(c*d - b*e)*(d + e*x)^6) - ((7*A*e*(2*c*d - b*e) - B*d*(2*c*d
+ 5*b*e))*(b*x + c*x^2)^(5/2))/(60*d^2*(c*d - b*e)^2*(d + e*x)^5) + (b^4*(24*A*c
^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*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])])/(1024*d^(9/2)*(c*d - b*
e)^(9/2))
_______________________________________________________________________________________
Rubi [A] time = 1.28773, antiderivative size = 402, 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
\[ -\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{512 d^4 (d+e x)^2 (c d-b e)^4}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )}{192 d^3 (d+e x)^4 (c d-b e)^3}+\frac{b^4 \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{60 d^2 (d+e x)^5 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^7,x]
[Out]
-(b^2*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d + (2*
c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(512*d^4*(c*d - b*e)^4*(d + e*x)^2) + ((24*A*c^
2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(b*d + (2*c*d - b*e)*x)*
(b*x + c*x^2)^(3/2))/(192*d^3*(c*d - b*e)^3*(d + e*x)^4) + ((B*d - A*e)*(b*x + c
*x^2)^(5/2))/(6*d*(c*d - b*e)*(d + e*x)^6) - ((7*A*e*(2*c*d - b*e) - B*d*(2*c*d
+ 5*b*e))*(b*x + c*x^2)^(5/2))/(60*d^2*(c*d - b*e)^2*(d + e*x)^5) + (b^4*(24*A*c
^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*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])])/(1024*d^(9/2)*(c*d - b*
e)^(9/2))
_______________________________________________________________________________________
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)**(3/2)/(e*x+d)**7,x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [A] time = 6.53928, size = 665, normalized size = 1.65 \[ \frac{(x (b+c x))^{3/2} \left (\frac{15 b^4 \left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{(b+c x)^{3/2} \sqrt{b e-c d}}+\frac{\sqrt{d} \sqrt{x} \left (16 d^3 (d+e x)^2 (c d-b e)^3 \left (A e \left (-3 b^2 e^2+152 b c d e-152 c^2 d^2\right )+B d \left (135 b^2 e^2-548 b c d e+416 c^2 d^2\right )\right )-8 d^2 (d+e x)^3 (c d-b e)^2 \left (A e \left (-7 b^3 e^3+6 b^2 c d e^2+24 b c^2 d^2 e-16 c^3 d^3\right )+B d \left (-5 b^3 e^3+438 b^2 c d e^2-888 b c^2 d^2 e+448 c^3 d^3\right )\right )+2 d (d+e x)^4 (c d-b e) \left (A e \left (-35 b^4 e^4+64 b^3 c d e^3-128 b c^3 d^3 e+64 c^4 d^4\right )+B d \left (-25 b^4 e^4+20 b^3 c d e^3+264 b^2 c^2 d^2 e^2-352 b c^3 d^3 e+128 c^4 d^4\right )\right )+(d+e x)^5 \left (A e \left (105 b^5 e^5-290 b^4 c d e^4+176 b^3 c^2 d^2 e^3+96 b^2 c^3 d^3 e^2-320 b c^4 d^4 e+128 c^5 d^5\right )+B d \left (75 b^5 e^5-130 b^4 c d e^4-80 b^3 c^2 d^2 e^3+816 b^2 c^3 d^3 e^2-832 b c^4 d^4 e+256 c^5 d^5\right )\right )+1280 d^5 (B d-A e) (c d-b e)^5-128 d^4 (d+e x) (c d-b e)^4 (13 A e (b e-2 c d)+B d (38 c d-25 b e))\right )}{e^4 (b+c x) (d+e x)^6}\right )}{7680 d^{9/2} x^{3/2} (c d-b e)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^7,x]
[Out]
((x*(b + c*x))^(3/2)*((Sqrt[d]*Sqrt[x]*(1280*d^5*(B*d - A*e)*(c*d - b*e)^5 - 128
*d^4*(c*d - b*e)^4*(B*d*(38*c*d - 25*b*e) + 13*A*e*(-2*c*d + b*e))*(d + e*x) + 1
6*d^3*(c*d - b*e)^3*(A*e*(-152*c^2*d^2 + 152*b*c*d*e - 3*b^2*e^2) + B*d*(416*c^2
*d^2 - 548*b*c*d*e + 135*b^2*e^2))*(d + e*x)^2 - 8*d^2*(c*d - b*e)^2*(A*e*(-16*c
^3*d^3 + 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 - 7*b^3*e^3) + B*d*(448*c^3*d^3 - 888*b*
c^2*d^2*e + 438*b^2*c*d*e^2 - 5*b^3*e^3))*(d + e*x)^3 + 2*d*(c*d - b*e)*(A*e*(64
*c^4*d^4 - 128*b*c^3*d^3*e + 64*b^3*c*d*e^3 - 35*b^4*e^4) + B*d*(128*c^4*d^4 - 3
52*b*c^3*d^3*e + 264*b^2*c^2*d^2*e^2 + 20*b^3*c*d*e^3 - 25*b^4*e^4))*(d + e*x)^4
+ (B*d*(256*c^5*d^5 - 832*b*c^4*d^4*e + 816*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^
3 - 130*b^4*c*d*e^4 + 75*b^5*e^5) + A*e*(128*c^5*d^5 - 320*b*c^4*d^4*e + 96*b^2*
c^3*d^3*e^2 + 176*b^3*c^2*d^2*e^3 - 290*b^4*c*d*e^4 + 105*b^5*e^5))*(d + e*x)^5)
)/(e^4*(b + c*x)*(d + e*x)^6) + (15*b^4*(24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) +
b^2*e*(5*B*d + 7*A*e))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*
x])])/(Sqrt[-(c*d) + b*e]*(b + c*x)^(3/2))))/(7680*d^(9/2)*(c*d - b*e)^4*x^(3/2)
)
_______________________________________________________________________________________
Maple [B] time = 0.069, size = 29243, normalized size = 72.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^7,x)
[Out]
result too large to display
_______________________________________________________________________________________
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)^(3/2)*(B*x + A)/(e*x + d)^7,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.327277, 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)^(3/2)*(B*x + A)/(e*x + d)^7,x, algorithm="fricas")
[Out]
[-1/15360*(2*(105*A*b^5*d^5*e^2 - 180*(B*b^4*c - 2*A*b^3*c^2)*d^7 + 15*(5*B*b^5
- 24*A*b^4*c)*d^6*e - (256*B*c^5*d^6*e + 105*A*b^5*e^7 - 64*(13*B*b*c^4 - 2*A*c^
5)*d^5*e^2 + 16*(51*B*b^2*c^3 - 20*A*b*c^4)*d^4*e^3 - 16*(5*B*b^3*c^2 - 6*A*b^2*
c^3)*d^3*e^4 - 2*(65*B*b^4*c - 88*A*b^3*c^2)*d^2*e^5 + 5*(15*B*b^5 - 58*A*b^4*c)
*d*e^6)*x^5 - (1536*B*c^5*d^7 + 595*A*b^5*d*e^6 - 256*(20*B*b*c^4 - 3*A*c^5)*d^6
*e + 64*(83*B*b^2*c^3 - 31*A*b*c^4)*d^5*e^2 - 8*(111*B*b^3*c^2 - 92*A*b^2*c^3)*d
^4*e^3 - 4*(185*B*b^4*c - 252*A*b^3*c^2)*d^3*e^4 + (425*B*b^5 - 1648*A*b^4*c)*d^
2*e^5)*x^4 - 6*(231*A*b^5*d^2*e^5 + 32*(11*B*b*c^4 + 10*A*c^5)*d^7 - 72*(19*B*b^
2*c^3 + 12*A*b*c^4)*d^6*e + 4*(475*B*b^3*c^2 + 102*A*b^2*c^3)*d^5*e^2 - 2*(437*B
*b^4*c - 186*A*b^3*c^2)*d^4*e^3 + 3*(55*B*b^5 - 214*A*b^4*c)*d^3*e^4)*x^3 - 2*(8
43*A*b^5*d^3*e^4 + 48*(B*b^2*c^3 + 30*A*b*c^4)*d^7 - 4*(89*B*b^3*c^2 + 1344*A*b^
2*c^3)*d^6*e + 4*(332*B*b^4*c + 1535*A*b^3*c^2)*d^5*e^2 - (495*B*b^5 + 3572*A*b^
4*c)*d^4*e^3)*x^2 + 5*(119*A*b^5*d^4*e^3 + 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^7 - 2*
(107*B*b^4*c - 228*A*b^3*c^2)*d^6*e + (85*B*b^5 - 422*A*b^4*c)*d^5*e^2)*x)*sqrt(
c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 15*(7*A*b^6*d^6*e^2 - 12*(B*b^5*c - 2*A*b^4*c
^2)*d^8 + (5*B*b^6 - 24*A*b^5*c)*d^7*e + (7*A*b^6*e^8 - 12*(B*b^5*c - 2*A*b^4*c^
2)*d^2*e^6 + (5*B*b^6 - 24*A*b^5*c)*d*e^7)*x^6 + 6*(7*A*b^6*d*e^7 - 12*(B*b^5*c
- 2*A*b^4*c^2)*d^3*e^5 + (5*B*b^6 - 24*A*b^5*c)*d^2*e^6)*x^5 + 15*(7*A*b^6*d^2*e
^6 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^4*e^4 + (5*B*b^6 - 24*A*b^5*c)*d^3*e^5)*x^4 +
20*(7*A*b^6*d^3*e^5 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^5*e^3 + (5*B*b^6 - 24*A*b^5*c
)*d^4*e^4)*x^3 + 15*(7*A*b^6*d^4*e^4 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^6*e^2 + (5*B
*b^6 - 24*A*b^5*c)*d^5*e^3)*x^2 + 6*(7*A*b^6*d^5*e^3 - 12*(B*b^5*c - 2*A*b^4*c^2
)*d^7*e + (5*B*b^6 - 24*A*b^5*c)*d^6*e^2)*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^14 - 4*
b*c^3*d^13*e + 6*b^2*c^2*d^12*e^2 - 4*b^3*c*d^11*e^3 + b^4*d^10*e^4 + (c^4*d^8*e
^6 - 4*b*c^3*d^7*e^7 + 6*b^2*c^2*d^6*e^8 - 4*b^3*c*d^5*e^9 + b^4*d^4*e^10)*x^6 +
6*(c^4*d^9*e^5 - 4*b*c^3*d^8*e^6 + 6*b^2*c^2*d^7*e^7 - 4*b^3*c*d^6*e^8 + b^4*d^
5*e^9)*x^5 + 15*(c^4*d^10*e^4 - 4*b*c^3*d^9*e^5 + 6*b^2*c^2*d^8*e^6 - 4*b^3*c*d^
7*e^7 + b^4*d^6*e^8)*x^4 + 20*(c^4*d^11*e^3 - 4*b*c^3*d^10*e^4 + 6*b^2*c^2*d^9*e
^5 - 4*b^3*c*d^8*e^6 + b^4*d^7*e^7)*x^3 + 15*(c^4*d^12*e^2 - 4*b*c^3*d^11*e^3 +
6*b^2*c^2*d^10*e^4 - 4*b^3*c*d^9*e^5 + b^4*d^8*e^6)*x^2 + 6*(c^4*d^13*e - 4*b*c^
3*d^12*e^2 + 6*b^2*c^2*d^11*e^3 - 4*b^3*c*d^10*e^4 + b^4*d^9*e^5)*x)*sqrt(c*d^2
- b*d*e)), -1/7680*((105*A*b^5*d^5*e^2 - 180*(B*b^4*c - 2*A*b^3*c^2)*d^7 + 15*(5
*B*b^5 - 24*A*b^4*c)*d^6*e - (256*B*c^5*d^6*e + 105*A*b^5*e^7 - 64*(13*B*b*c^4 -
2*A*c^5)*d^5*e^2 + 16*(51*B*b^2*c^3 - 20*A*b*c^4)*d^4*e^3 - 16*(5*B*b^3*c^2 - 6
*A*b^2*c^3)*d^3*e^4 - 2*(65*B*b^4*c - 88*A*b^3*c^2)*d^2*e^5 + 5*(15*B*b^5 - 58*A
*b^4*c)*d*e^6)*x^5 - (1536*B*c^5*d^7 + 595*A*b^5*d*e^6 - 256*(20*B*b*c^4 - 3*A*c
^5)*d^6*e + 64*(83*B*b^2*c^3 - 31*A*b*c^4)*d^5*e^2 - 8*(111*B*b^3*c^2 - 92*A*b^2
*c^3)*d^4*e^3 - 4*(185*B*b^4*c - 252*A*b^3*c^2)*d^3*e^4 + (425*B*b^5 - 1648*A*b^
4*c)*d^2*e^5)*x^4 - 6*(231*A*b^5*d^2*e^5 + 32*(11*B*b*c^4 + 10*A*c^5)*d^7 - 72*(
19*B*b^2*c^3 + 12*A*b*c^4)*d^6*e + 4*(475*B*b^3*c^2 + 102*A*b^2*c^3)*d^5*e^2 - 2
*(437*B*b^4*c - 186*A*b^3*c^2)*d^4*e^3 + 3*(55*B*b^5 - 214*A*b^4*c)*d^3*e^4)*x^3
- 2*(843*A*b^5*d^3*e^4 + 48*(B*b^2*c^3 + 30*A*b*c^4)*d^7 - 4*(89*B*b^3*c^2 + 13
44*A*b^2*c^3)*d^6*e + 4*(332*B*b^4*c + 1535*A*b^3*c^2)*d^5*e^2 - (495*B*b^5 + 35
72*A*b^4*c)*d^4*e^3)*x^2 + 5*(119*A*b^5*d^4*e^3 + 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d
^7 - 2*(107*B*b^4*c - 228*A*b^3*c^2)*d^6*e + (85*B*b^5 - 422*A*b^4*c)*d^5*e^2)*x
)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 15*(7*A*b^6*d^6*e^2 - 12*(B*b^5*c - 2
*A*b^4*c^2)*d^8 + (5*B*b^6 - 24*A*b^5*c)*d^7*e + (7*A*b^6*e^8 - 12*(B*b^5*c - 2*
A*b^4*c^2)*d^2*e^6 + (5*B*b^6 - 24*A*b^5*c)*d*e^7)*x^6 + 6*(7*A*b^6*d*e^7 - 12*(
B*b^5*c - 2*A*b^4*c^2)*d^3*e^5 + (5*B*b^6 - 24*A*b^5*c)*d^2*e^6)*x^5 + 15*(7*A*b
^6*d^2*e^6 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^4*e^4 + (5*B*b^6 - 24*A*b^5*c)*d^3*e^5
)*x^4 + 20*(7*A*b^6*d^3*e^5 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^5*e^3 + (5*B*b^6 - 24
*A*b^5*c)*d^4*e^4)*x^3 + 15*(7*A*b^6*d^4*e^4 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^6*e^
2 + (5*B*b^6 - 24*A*b^5*c)*d^5*e^3)*x^2 + 6*(7*A*b^6*d^5*e^3 - 12*(B*b^5*c - 2*A
*b^4*c^2)*d^7*e + (5*B*b^6 - 24*A*b^5*c)*d^6*e^2)*x)*arctan(-sqrt(-c*d^2 + b*d*e
)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c^4*d^14 - 4*b*c^3*d^13*e + 6*b^2*c^2*d^
12*e^2 - 4*b^3*c*d^11*e^3 + b^4*d^10*e^4 + (c^4*d^8*e^6 - 4*b*c^3*d^7*e^7 + 6*b^
2*c^2*d^6*e^8 - 4*b^3*c*d^5*e^9 + b^4*d^4*e^10)*x^6 + 6*(c^4*d^9*e^5 - 4*b*c^3*d
^8*e^6 + 6*b^2*c^2*d^7*e^7 - 4*b^3*c*d^6*e^8 + b^4*d^5*e^9)*x^5 + 15*(c^4*d^10*e
^4 - 4*b*c^3*d^9*e^5 + 6*b^2*c^2*d^8*e^6 - 4*b^3*c*d^7*e^7 + b^4*d^6*e^8)*x^4 +
20*(c^4*d^11*e^3 - 4*b*c^3*d^10*e^4 + 6*b^2*c^2*d^9*e^5 - 4*b^3*c*d^8*e^6 + b^4*
d^7*e^7)*x^3 + 15*(c^4*d^12*e^2 - 4*b*c^3*d^11*e^3 + 6*b^2*c^2*d^10*e^4 - 4*b^3*
c*d^9*e^5 + b^4*d^8*e^6)*x^2 + 6*(c^4*d^13*e - 4*b*c^3*d^12*e^2 + 6*b^2*c^2*d^11
*e^3 - 4*b^3*c*d^10*e^4 + b^4*d^9*e^5)*x)*sqrt(-c*d^2 + b*d*e))]
_______________________________________________________________________________________
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((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**7,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.630709, 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)^(3/2)*(B*x + A)/(e*x + d)^7,x, algorithm="giac")
[Out]
sage0*x