3.2258 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\)
Optimal. Leaf size=372 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt{d+e x}}-\frac{5 c \sqrt{2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2} \]
[Out]
(5*c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(
4*e^2*Sqrt[d + e*x]) + (5*c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)) + ((3*c*e*f - 11*
c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(4*e^2*(2*c*d - b*
e)*(d + e*x)^(7/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/
(2*e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) - (5*c*Sqrt[2*c*d - b*e]*(3*c*e*f - 11*c*
d*g + 4*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b
*e]*Sqrt[d + e*x])])/(4*e^2)
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Rubi [A] time = 1.31184, antiderivative size = 372, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109
\[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt{d+e x}}-\frac{5 c \sqrt{2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(11/2),x]
[Out]
(5*c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(
4*e^2*Sqrt[d + e*x]) + (5*c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^
2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)) + ((3*c*e*f - 11*
c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(4*e^2*(2*c*d - b*
e)*(d + e*x)^(7/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/
(2*e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) - (5*c*Sqrt[2*c*d - b*e]*(3*c*e*f - 11*c*
d*g + 4*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b
*e]*Sqrt[d + e*x])])/(4*e^2)
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Rubi in Sympy [A] time = 159.657, size = 352, normalized size = 0.95 \[ - \frac{5 c \sqrt{b e - 2 c d} \left (4 b e g - 11 c d g + 3 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{4 e^{2}} + \frac{5 c \left (4 b e g - 11 c d g + 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 e^{2} \sqrt{d + e x}} - \frac{5 c \left (4 b e g - 11 c d g + 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{12 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )} - \frac{\left (4 b e g - 11 c d g + 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{4 e^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{2 e^{2} \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(11/2),x)
[Out]
-5*c*sqrt(b*e - 2*c*d)*(4*b*e*g - 11*c*d*g + 3*c*e*f)*atan(sqrt(-b*e**2*x - c*e*
*2*x**2 + d*(-b*e + c*d))/(sqrt(d + e*x)*sqrt(b*e - 2*c*d)))/(4*e**2) + 5*c*(4*b
*e*g - 11*c*d*g + 3*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(4*e**
2*sqrt(d + e*x)) - 5*c*(4*b*e*g - 11*c*d*g + 3*c*e*f)*(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d))**(3/2)/(12*e**2*(d + e*x)**(3/2)*(b*e - 2*c*d)) - (4*b*e*g - 11
*c*d*g + 3*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(4*e**2*(d +
e*x)**(7/2)*(b*e - 2*c*d)) - (d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c
*d))**(7/2)/(2*e**2*(d + e*x)**(11/2)*(b*e - 2*c*d))
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Mathematica [A] time = 2.26952, size = 238, normalized size = 0.64 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{15 c \sqrt{2 c d-b e} (-4 b e g+11 c d g-3 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(c (d-e x)-b e)^{5/2}}-\frac{-8 c (d+e x)^2 (7 b e g-16 c d g+3 c e f)+3 (d+e x) (2 c d-b e) (-4 b e g+17 c d g-9 c e f)+6 (b e-2 c d)^2 (e f-d g)-8 c^2 e g x (d+e x)^2}{(d+e x)^2 (b e-c d+c e x)^2}\right )}{12 e^2 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(11/2),x]
[Out]
(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(-((6*(-2*c*d + b*e)^2*(e*f - d*g) + 3
*(2*c*d - b*e)*(-9*c*e*f + 17*c*d*g - 4*b*e*g)*(d + e*x) - 8*c*(3*c*e*f - 16*c*d
*g + 7*b*e*g)*(d + e*x)^2 - 8*c^2*e*g*x*(d + e*x)^2)/((d + e*x)^2*(-(c*d) + b*e
+ c*e*x)^2)) + (15*c*Sqrt[2*c*d - b*e]*(-3*c*e*f + 11*c*d*g - 4*b*e*g)*ArcTanh[S
qrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]])/(-(b*e) + c*(d - e*x))^(5/2)))/(12*e^
2*(d + e*x)^(5/2))
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Maple [B] time = 0.044, size = 1189, normalized size = 3.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(11/2),x)
[Out]
-1/12*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(6*b^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*
e*x-b*e+c*d)^(1/2)+120*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b^2*c*
d*e^3*g+90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^2*d*e^3*f+60*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b^2*c*e^4*g+45*arctan((-c*e*
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*e^4*f+206*c^2*d^3*g*(b*e-2*c*d)^(1
/2)*(-c*e*x-b*e+c*d)^(1/2)-90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c
^3*d^3*e*f+330*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4*g+45*arc
tan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*f+60*arctan((-c*e*x-
b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d^2*e^2*g-8*x^3*c^2*e^3*g*(-c*e*x-b*e+c*
d)^(1/2)*(b*e-2*c*d)^(1/2)+27*x*b*c*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/
2)+350*x*c^2*d^2*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-102*x*c^2*d*e^2*f*
(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-107*b*c*d^2*e*g*(b*e-2*c*d)^(1/2)*(-c*e
*x-b*e+c*d)^(1/2)+3*b*c*d*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-285*arc
tan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*d*e^3*g-570*arctan((-c*e
*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^2*d^2*e^2*g-56*x^2*b*c*e^3*g*(b*e-2*c
*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+112*x^2*c^2*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b
*e+c*d)^(1/2)-24*x^2*c^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+12*x*b^2
*e^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+6*b^2*d*e^2*g*(b*e-2*c*d)^(1/2)*
(-c*e*x-b*e+c*d)^(1/2)-54*c^2*d^2*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+3
30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d^2*e^2*g-90*arctan(
(-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d*e^3*f+660*arctan((-c*e*x-b*e
+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^3*e*g-180*arctan((-c*e*x-b*e+c*d)^(1/2)/(
b*e-2*c*d)^(1/2))*x*c^3*d^2*e^2*f-285*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^
(1/2))*b*c^2*d^3*e*g-187*x*b*c*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2))
/(e*x+d)^(5/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)
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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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(11/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.336042, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(11/2),x, algorithm="fricas")
[Out]
[-1/24*(16*c^3*e^4*g*x^4 + 16*(3*c^3*e^4*f - (15*c^3*d*e^3 - 8*b*c^2*e^4)*g)*x^3
- 15*(3*c^2*d*e*f - (11*c^2*d^2 - 4*b*c*d*e)*g + (3*c^2*e^2*f - (11*c^2*d*e - 4
*b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqr
t(e*x + d)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x + 2*sqrt(-c
*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2*x^2 +
2*d*e*x + d^2)) + 2*(3*(26*c^3*d*e^3 - b*c^2*e^4)*f - (238*c^3*d^2*e^2 - 19*b*c^
2*d*e^3 - 44*b^2*c*e^4)*g)*x^2 - 6*(18*c^3*d^3*e - 19*b*c^2*d^2*e^2 - b^2*c*d*e^
3 + 2*b^3*e^4)*f + 2*(206*c^3*d^4 - 313*b*c^2*d^3*e + 113*b^2*c*d^2*e^2 - 6*b^3*
d*e^3)*g - 2*(3*(16*c^3*d^2*e^2 - 42*b*c^2*d*e^3 + 11*b^2*c*e^4)*f - (144*c^3*d^
3*e - 430*b*c^2*d^2*e^2 + 193*b^2*c*d*e^3 - 12*b^3*e^4)*g)*x)/(sqrt(-c*e^2*x^2 -
b*e^2*x + c*d^2 - b*d*e)*(e^3*x + d*e^2)*sqrt(e*x + d)), -1/12*(8*c^3*e^4*g*x^4
+ 8*(3*c^3*e^4*f - (15*c^3*d*e^3 - 8*b*c^2*e^4)*g)*x^3 + 15*(3*c^2*d*e*f - (11*
c^2*d^2 - 4*b*c*d*e)*g + (3*c^2*e^2*f - (11*c^2*d*e - 4*b*c*e^2)*g)*x)*sqrt(-c*e
^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)*arctan(-sqrt(
-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*d - b*e)*sqrt(e*x + d)/((c*e^2*x^2 +
b*e^2*x - c*d^2 + b*d*e)*sqrt(-2*c*d + b*e))) + (3*(26*c^3*d*e^3 - b*c^2*e^4)*f
- (238*c^3*d^2*e^2 - 19*b*c^2*d*e^3 - 44*b^2*c*e^4)*g)*x^2 - 3*(18*c^3*d^3*e - 1
9*b*c^2*d^2*e^2 - b^2*c*d*e^3 + 2*b^3*e^4)*f + (206*c^3*d^4 - 313*b*c^2*d^3*e +
113*b^2*c*d^2*e^2 - 6*b^3*d*e^3)*g - (3*(16*c^3*d^2*e^2 - 42*b*c^2*d*e^3 + 11*b^
2*c*e^4)*f - (144*c^3*d^3*e - 430*b*c^2*d^2*e^2 + 193*b^2*c*d*e^3 - 12*b^3*e^4)*
g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e^3*x + d*e^2)*sqrt(e*x + d))
]
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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((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(11/2),x)
[Out]
Timed out
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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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(11/2),x, algorithm="giac")
[Out]
Timed out