3.2267 \(\int \frac{f+g x}{(d+e x)^{5/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=233 \[ -\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{c (-4 b e g+5 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{5/2}} \]

[Out]

-((e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(2*e^2*(2*c*d - b*e)*(d
+ e*x)^(5/2)) - ((3*c*e*f + 5*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*(2*c*d - b*e)^2*(d + e*x)^(3/2)) - (c*(3*c*e*f + 5*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*(2*c*d - b*e)^(5/2))

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Rubi [A]  time = 0.819846, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{c (-4 b e g+5 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(2*e^2*(2*c*d - b*e)*(d
+ e*x)^(5/2)) - ((3*c*e*f + 5*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*(2*c*d - b*e)^2*(d + e*x)^(3/2)) - (c*(3*c*e*f + 5*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*(2*c*d - b*e)^(5/2))

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Rubi in Sympy [A]  time = 93.1591, size = 214, normalized size = 0.92 \[ - \frac{c \left (4 b e g - 5 c d g - 3 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{4 e^{2} \left (b e - 2 c d\right )^{\frac{5}{2}}} + \frac{\left (4 b e g - 5 c d g - 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )^{2}} - \frac{\left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{2 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)/(e*x+d)**(5/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

-c*(4*b*e*g - 5*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*(b*e - 2*c*d)**(5/2)) + (4*b*e*g
- 5*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*(d
+ e*x)**(3/2)*(b*e - 2*c*d)**2) - (d*g - e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(
-b*e + c*d))/(2*e**2*(d + e*x)**(5/2)*(b*e - 2*c*d))

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Mathematica [A]  time = 1.56842, size = 194, normalized size = 0.83 \[ -\frac{\sqrt{d+e x} \left (\frac{(c (d-e x)-b e) \left (c \left (d^2 g+d e (7 f+5 g x)+3 e^2 f x\right )-2 b e (d g+e (f+2 g x))\right )}{(d+e x)^2 (b e-2 c d)^2}+\frac{c \sqrt{c (d-e x)-b e} (-4 b e g+5 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{5/2}}\right )}{4 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*(((-(b*e) + c*(d - e*x))*(-2*b*e*(d*g + e*(f + 2*g*x)) + c*(d^2*
g + 3*e^2*f*x + d*e*(7*f + 5*g*x))))/((-2*c*d + b*e)^2*(d + e*x)^2) + (c*(3*c*e*
f + 5*c*d*g - 4*b*e*g)*Sqrt[-(b*e) + c*(d - e*x)]*ArcTanh[Sqrt[c*d - b*e - c*e*x
]/Sqrt[2*c*d - b*e]])/(2*c*d - b*e)^(5/2)))/(4*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d
- e*x))])

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Maple [B]  time = 0.042, size = 630, normalized size = 2.7 \[ -{\frac{1}{4\,{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}bc{e}^{3}g-5\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}d{e}^{2}g-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}{e}^{3}f+8\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbcd{e}^{2}g-10\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}{d}^{2}eg-6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}d{e}^{2}f+4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{d}^{2}eg-5\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{3}g-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{2}ef-4\,xb{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+5\,xcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+3\,xc{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-2\,bdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-2\,b{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+c{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+7\,cdef\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}} \left ( be-2\,cd \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-1/4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*x^2*b*c*e^3*g-5*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))
*x^2*c^2*d*e^2*g-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^2*e^3*
f+8*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c*d*e^2*g-10*arctan((-c
*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d^2*e*g-6*arctan((-c*e*x-b*e+c*d)^(
1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d*e^2*f+4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d
)^(1/2))*b*c*d^2*e*g-5*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*
g-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f-4*x*b*e^2*g*(-c
*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+5*x*c*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*
c*d)^(1/2)+3*x*c*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-2*b*d*e*g*(-c*e*
x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-2*b*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^
(1/2)+c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+7*c*d*e*f*(-c*e*x-b*e+c*d
)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^(5/2)/(b*e-2*c*d)^(5/2)/e^2/(-c*e*x-b*e+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((g*x + f)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294725, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[-1/8*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*((7*c*d*e
- 2*b*e^2)*f + (c*d^2 - 2*b*d*e)*g + (3*c*e^2*f + (5*c*d*e - 4*b*e^2)*g)*x)*sqrt
(e*x + d) + (3*c^2*d^3*e*f + (3*c^2*e^4*f + (5*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3
*(3*c^2*d*e^3*f + (5*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (5*c^2*d^4 - 4*b*c*d^3*
e)*g + 3*(3*c^2*d^2*e^2*f + (5*c^2*d^3*e - 4*b*c*d^2*e^2)*g)*x)*log((2*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 - 3*c
*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x)*sqrt(2*c*d - b*e))/(e^2*x^2 + 2*d*e*x + d^
2)))/((4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*e^
6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c*d^2*e^5 + b^2*d*e^6)*x^2 + 3*(4*c^2*
d^4*e^3 - 4*b*c*d^3*e^4 + b^2*d^2*e^5)*x)*sqrt(2*c*d - b*e)), -1/4*(sqrt(-c*e^2*
x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*((7*c*d*e - 2*b*e^2)*f + (c*d^
2 - 2*b*d*e)*g + (3*c*e^2*f + (5*c*d*e - 4*b*e^2)*g)*x)*sqrt(e*x + d) - (3*c^2*d
^3*e*f + (3*c^2*e^4*f + (5*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(3*c^2*d*e^3*f + (5
*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (5*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*
e^2*f + (5*c^2*d^3*e - 4*b*c*d^2*e^2)*g)*x)*arctan(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)/(c*e^2*x^2 + b*e^2*x - c*d^2 + b*
d*e)))/((4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*
e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c*d^2*e^5 + b^2*d*e^6)*x^2 + 3*(4*c^
2*d^4*e^3 - 4*b*c*d^3*e^4 + b^2*d^2*e^5)*x)*sqrt(-2*c*d + b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f + g x}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)/(e*x+d)**(5/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 22.2542, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

sage0*x