3.2237 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{c (-4 b e g+7 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]

[Out]

-((c*e*f + 7*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)*(d + e*x)^(3/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(3/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) + (c*(c*e*f + 7*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)^(3/2))

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Rubi [A]  time = 0.849592, antiderivative size = 231, 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 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{c (-4 b e g+7 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(7/2),x]

[Out]

-((c*e*f + 7*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)*(d + e*x)^(3/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(3/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) + (c*(c*e*f + 7*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)^(3/2))

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Rubi in Sympy [A]  time = 90.4418, size = 211, normalized size = 0.91 \[ - \frac{c \left (4 b e g - 7 c d g - c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{4 e^{2} \left (b e - 2 c d\right )^{\frac{3}{2}}} - \frac{\left (4 b e g - 7 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{2 e^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )} \]

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)**(1/2)/(e*x+d)**(7/2),x)

[Out]

-c*(4*b*e*g - 7*c*d*g - 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)**(3/2)) - (4*b*e*g -
7*c*d*g - 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)) - (d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)
)**(3/2)/(2*e**2*(d + e*x)**(7/2)*(b*e - 2*c*d))

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Mathematica [A]  time = 0.730292, size = 170, normalized size = 0.74 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{4 b e g-9 c d g+c e f}{(d+e x) (2 c d-b e)}+\frac{c (-4 b e g+7 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{3/2} \sqrt{c (d-e x)-b e}}+\frac{2 (d g-e f)}{(d+e x)^2}\right )}{4 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(7/2),x]

[Out]

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

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Maple [B]  time = 0.035, size = 630, normalized size = 2.7 \[ -{\frac{1}{4\,{e}^{2}} \left ( 4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}bc{e}^{3}g-7\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}d{e}^{2}g-\arctan \left ({1\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ){x}^{2}{c}^{2}{e}^{3}f+8\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbcd{e}^{2}g-14\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}{d}^{2}eg-2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}d{e}^{2}f+4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{d}^{2}eg-7\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{3}g-\arctan \left ({1\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ){c}^{2}{d}^{2}ef+4\,xb{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-9\,xcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+xc{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+2\,bdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+2\,b{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-5\,c{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-3\,cdef\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( be-2\,cd \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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)^(1/2)/(e*x+d)^(7/2),x)

[Out]

-1/4*(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-7*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-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-14*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*
c^2*d^2*e*g-2*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*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d^2*e*g-7*arctan((-c*e*x-b*e
+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g-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)-9*x*c
*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+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)-5*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2
*c*d)^(1/2)-3*c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)/(b*e-2*c*d)^(3/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)/(e*x+d)^(5/2
)

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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(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300842, 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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(7/2),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)*((3*c*d*e
- 2*b*e^2)*f + (5*c*d^2 - 2*b*d*e)*g - (c*e^2*f - (9*c*d*e - 4*b*e^2)*g)*x)*sqrt
(e*x + d) - (c^2*d^3*e*f + (c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^
2*d*e^3*f + (7*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c*d^3*e)*g +
3*(c^2*d^2*e^2*f + (7*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)))/((2
*c*d^4*e^2 - b*d^3*e^3 + (2*c*d*e^5 - b*e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)*x^2
+ 3*(2*c*d^3*e^3 - b*d^2*e^4)*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)*((3*c*d*e - 2*b*e^2)*f + (5*c*d^2 - 2*
b*d*e)*g - (c*e^2*f - (9*c*d*e - 4*b*e^2)*g)*x)*sqrt(e*x + d) + (c^2*d^3*e*f + (
c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^2*d*e^3*f + (7*c^2*d^2*e^2 -
4*b*c*d*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(c^2*d^2*e^2*f + (7*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)*sq
rt(-2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)))/((2*c*d^4
*e^2 - b*d^3*e^3 + (2*c*d*e^5 - b*e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)*x^2 + 3*(
2*c*d^3*e^3 - b*d^2*e^4)*x)*sqrt(-2*c*d + b*e))]

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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)**(1/2)/(e*x+d)**(7/2),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(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Timed out