3.2256 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e)^2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]

[Out]

(2*(2*c*d - b*e)^2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*S
qrt[d + e*x]) + (2*(2*c*d - b*e)*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) + (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(5/2))/(5*e^2*(d + e*x)^(5/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(7*c*e^2*(d + e*x)^(7/2)) - (2*(2*c*d - b*e)^(5/2)*(e*f - d*g)*ArcTa
nh[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])])
/e^2

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Rubi [A]  time = 1.24924, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e)^2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]

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)^(7/2),x]

[Out]

(2*(2*c*d - b*e)^2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*S
qrt[d + e*x]) + (2*(2*c*d - b*e)*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) + (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(5/2))/(5*e^2*(d + e*x)^(5/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(7*c*e^2*(d + e*x)^(7/2)) - (2*(2*c*d - b*e)^(5/2)*(e*f - d*g)*ArcTa
nh[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])])
/e^2

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Rubi in Sympy [A]  time = 146.255, size = 282, normalized size = 0.89 \[ \frac{2 \left (b e - 2 c d\right )^{\frac{5}{2}} \left (d g - e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2}} - \frac{2 \left (b e - 2 c d\right )^{2} \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \sqrt{d + e x}} + \frac{2 \left (b e - 2 c d\right ) \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{5 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{7 c e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(7/2),x)

[Out]

2*(b*e - 2*c*d)**(5/2)*(d*g - 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)))/e**2 - 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))/(e**2*sqrt(d + e*x)) + 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)/(3*e**2*(d +
e*x)**(3/2)) - 2*(d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/
(5*e**2*(d + e*x)**(5/2)) - 2*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2
)/(7*c*e**2*(d + e*x)**(7/2))

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Mathematica [A]  time = 1.61551, size = 255, normalized size = 0.81 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{15 b^3 e^3 g+b^2 c e^2 (-206 d g+161 e f+45 e g x)+b c^2 e \left (612 d^2 g-d e (567 f+167 g x)+e^2 x (77 f+45 g x)\right )+c^3 \left (-526 d^3 g+d^2 e (511 f+157 g x)-2 d e^2 x (56 f+33 g x)+3 e^3 x^2 (7 f+5 g x)\right )}{c (b e-c d+c e x)^2}+\frac{105 (2 c d-b e)^{5/2} (d g-e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(c (d-e x)-b e)^{5/2}}\right )}{105 e^2 (d+e x)^{5/2}} \]

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)^(7/2),x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((15*b^3*e^3*g + b^2*c*e^2*(161*e*f
- 206*d*g + 45*e*g*x) + c^3*(-526*d^3*g + 3*e^3*x^2*(7*f + 5*g*x) - 2*d*e^2*x*(5
6*f + 33*g*x) + d^2*e*(511*f + 157*g*x)) + b*c^2*e*(612*d^2*g + e^2*x*(77*f + 45
*g*x) - d*e*(567*f + 167*g*x)))/(c*(-(c*d) + b*e + c*e*x)^2) + (105*(2*c*d - b*e
)^(5/2)*(-(e*f) + d*g)*ArcTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]])/(-(b*
e) + c*(d - e*x))^(5/2)))/(105*e^2*(d + e*x)^(5/2))

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Maple [B]  time = 0.03, size = 956, normalized size = 3. \[{\frac{2}{105\,c{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 15\,{x}^{3}{c}^{3}{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+45\,{x}^{2}b{c}^{2}{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-66\,{x}^{2}{c}^{3}d{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+21\,{x}^{2}{c}^{3}{e}^{3}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+105\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{3}cd{e}^{3}g-105\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{3}c{e}^{4}f-630\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}{c}^{2}{d}^{2}{e}^{2}g+630\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}{c}^{2}d{e}^{3}f+1260\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{3}{d}^{3}eg-1260\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{3}{d}^{2}{e}^{2}f-840\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{4}{d}^{4}g+840\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{4}{d}^{3}ef+45\,x{b}^{2}c{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-167\,xb{c}^{2}d{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+77\,xb{c}^{2}{e}^{3}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+157\,x{c}^{3}{d}^{2}eg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-112\,x{c}^{3}d{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+15\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{3}{e}^{3}g-206\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{2}cd{e}^{2}g+161\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{2}c{e}^{3}f+612\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}b{c}^{2}{d}^{2}eg-567\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}b{c}^{2}d{e}^{2}f-526\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{c}^{3}{d}^{3}g+511\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{c}^{3}{d}^{2}ef \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \]

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)^(7/2),x)

[Out]

2/105*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(15*x^3*c^3*e^3*g*(-c*e*x-b*e+c*d)^
(1/2)*(b*e-2*c*d)^(1/2)+45*x^2*b*c^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1
/2)-66*x^2*c^3*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+21*x^2*c^3*e^3*f
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))*b^3*c*d*e^3*g-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*b^3*c*e^4*f-630*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d^2*e
^2*g+630*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d*e^3*f+1260*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^3*e*g-1260*arctan((-c*e*
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^2*e^2*f-840*arctan((-c*e*x-b*e+c*d)^
(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*g+840*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)
^(1/2))*c^4*d^3*e*f+45*x*b^2*c*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-16
7*x*b*c^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+77*x*b*c^2*e^3*f*(-c*
e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+157*x*c^3*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b
*e-2*c*d)^(1/2)-112*x*c^3*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+15*(b
*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b^3*e^3*g-206*(b*e-2*c*d)^(1/2)*(-c*e*x-b
*e+c*d)^(1/2)*b^2*c*d*e^2*g+161*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b^2*c*e
^3*f+612*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b*c^2*d^2*e*g-567*(b*e-2*c*d)^
(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b*c^2*d*e^2*f-526*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d
)^(1/2)*c^3*d^3*g+511*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*c^3*d^2*e*f)/(e*x
+d)^(1/2)/(-c*e*x-b*e+c*d)^(1/2)/c/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)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312666, 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)^(7/2),x, algorithm="fricas")

[Out]

[-1/105*(30*c^4*e^5*g*x^5 + 6*(7*c^4*e^5*f - 2*(11*c^4*d*e^4 - 10*b*c^3*e^5)*g)*
x^4 - 4*(7*(8*c^4*d*e^4 - 7*b*c^3*e^5)*f - (71*c^4*d^2*e^3 - 109*b*c^3*d*e^4 + 4
5*b^2*c^2*e^5)*g)*x^3 + 105*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*
d - b*e)*((4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f - (4*c^3*d^3 - 4*b*c^2*d^2
*e + b^2*c*d*e^2)*g)*sqrt(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)*sq
rt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(7*(35*c^4*d^2*e^3 - 47*b*c^3*d*e^4
+ 17*b^2*c^2*e^5)*f - (230*c^4*d^3*e^2 - 329*b*c^3*d^2*e^3 + 164*b^2*c^2*d*e^4 -
30*b^3*c*e^5)*g)*x^2 - 14*(73*c^4*d^4*e - 154*b*c^3*d^3*e^2 + 104*b^2*c^2*d^2*e
^3 - 23*b^3*c*d*e^4)*f + 2*(526*c^4*d^5 - 1138*b*c^3*d^4*e + 818*b^2*c^2*d^3*e^2
- 221*b^3*c*d^2*e^3 + 15*b^4*d*e^4)*g + 2*(7*(16*c^4*d^3*e^2 + 46*b*c^3*d^2*e^3
- 70*b^2*c^2*d*e^4 + 23*b^3*c*e^5)*f - (157*c^4*d^4*e + 202*b*c^3*d^3*e^2 - 400
*b^2*c^2*d^2*e^3 + 161*b^3*c*d*e^4 - 15*b^4*e^5)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*
x + c*d^2 - b*d*e)*sqrt(e*x + d)*c*e^2), -2/105*(15*c^4*e^5*g*x^5 + 3*(7*c^4*e^5
*f - 2*(11*c^4*d*e^4 - 10*b*c^3*e^5)*g)*x^4 - 2*(7*(8*c^4*d*e^4 - 7*b*c^3*e^5)*f
- (71*c^4*d^2*e^3 - 109*b*c^3*d*e^4 + 45*b^2*c^2*e^5)*g)*x^3 + 105*sqrt(-c*e^2*
x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*((4*c^3*d^2*e - 4*b*c^2*d*e^2
+ b^2*c*e^3)*f - (4*c^3*d^3 - 4*b*c^2*d^2*e + b^2*c*d*e^2)*g)*sqrt(e*x + d)*arct
an(-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))) + 2*(7*(35*c^4*d^2*e^3 -
47*b*c^3*d*e^4 + 17*b^2*c^2*e^5)*f - (230*c^4*d^3*e^2 - 329*b*c^3*d^2*e^3 + 164*
b^2*c^2*d*e^4 - 30*b^3*c*e^5)*g)*x^2 - 7*(73*c^4*d^4*e - 154*b*c^3*d^3*e^2 + 104
*b^2*c^2*d^2*e^3 - 23*b^3*c*d*e^4)*f + (526*c^4*d^5 - 1138*b*c^3*d^4*e + 818*b^2
*c^2*d^3*e^2 - 221*b^3*c*d^2*e^3 + 15*b^4*d*e^4)*g + (7*(16*c^4*d^3*e^2 + 46*b*c
^3*d^2*e^3 - 70*b^2*c^2*d*e^4 + 23*b^3*c*e^5)*f - (157*c^4*d^4*e + 202*b*c^3*d^3
*e^2 - 400*b^2*c^2*d^2*e^3 + 161*b^3*c*d*e^4 - 15*b^4*e^5)*g)*x)/(sqrt(-c*e^2*x^
2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*c*e^2)]

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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)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(7/2),x, algorithm="giac")

[Out]

Timed out