3.2281 \(\int \frac{(d+e x)^{3/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=221 \[ \frac{2 (d+e x)^{3/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} (e f-d g)}{e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 (2 c d-b e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.754745, antiderivative size = 221, 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 \[ \frac{2 (d+e x)^{3/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} (e f-d g)}{e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 (2 c d-b e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 93.6857, size = 197, normalized size = 0.89 \[ - \frac{2 \sqrt{d + e x} \left (d g - e f\right )}{e^{2} \left (b e - 2 c d\right )^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{2 \left (d g - e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2} \left (b e - 2 c d\right )^{\frac{5}{2}}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 1.24105, size = 187, normalized size = 0.85 \[ \frac{2 (d+e x)^{5/2} \left (\frac{3 (d g-e f) (c (d-e x)-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{5/2}}-\frac{(c (d-e x)-b e) \left (-b^2 e^2 g+4 b c e^2 f+c^2 \left (d^2 g-d e (5 f+3 g x)+3 e^2 f x\right )\right )}{c (b e-2 c d)^2}\right )}{3 e^2 ((d+e x) (c (d-e x)-b e))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.77, size = 485, normalized size = 2.2 \[{\frac{2}{3\,c{e}^{2} \left ( cex+be-cd \right ) ^{2}} \left ( 3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}deg\sqrt{-cex-be+cd}-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}{e}^{2}f\sqrt{-cex-be+cd}+3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bcdeg\sqrt{-cex-be+cd}-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{e}^{2}f\sqrt{-cex-be+cd}-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{2}g\sqrt{-cex-be+cd}+3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}def\sqrt{-cex-be+cd}+3\,\sqrt{be-2\,cd}x{c}^{2}deg-3\,\sqrt{be-2\,cd}x{c}^{2}{e}^{2}f+\sqrt{be-2\,cd}{b}^{2}{e}^{2}g-4\,\sqrt{be-2\,cd}bc{e}^{2}f-\sqrt{be-2\,cd}{c}^{2}{d}^{2}g+5\,\sqrt{be-2\,cd}{c}^{2}def \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( be-2\,cd \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298853, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*((5*c^2*d*e
- 4*b*c*e^2)*f - (c^2*d^2 - b^2*e^2)*g - 3*(c^2*e^2*f - c^2*d*e*g)*x)*sqrt(e*x
+ d) - 3*((c^3*e^4*f - c^3*d*e^3*g)*x^3 - ((c^3*d*e^3 - 2*b*c^2*e^4)*f - (c^3*d^
2*e^2 - 2*b*c^2*d*e^3)*g)*x^2 + (c^3*d^3*e - 2*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f -
(c^3*d^4 - 2*b*c^2*d^3*e + b^2*c*d^2*e^2)*g - ((c^3*d^2*e^2 - b^2*c*e^4)*f - (c^
3*d^3*e - b^2*c*d*e^3)*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^5*d^5*e^2 - 12*b*c^4*d^4
*e^3 + 13*b^2*c^3*d^3*e^4 - 6*b^3*c^2*d^2*e^5 + b^4*c*d*e^6 + (4*c^5*d^2*e^5 - 4
*b*c^4*d*e^6 + b^2*c^3*e^7)*x^3 - (4*c^5*d^3*e^4 - 12*b*c^4*d^2*e^5 + 9*b^2*c^3*
d*e^6 - 2*b^3*c^2*e^7)*x^2 - (4*c^5*d^4*e^3 - 4*b*c^4*d^3*e^4 - 3*b^2*c^3*d^2*e^
5 + 4*b^3*c^2*d*e^6 - b^4*c*e^7)*x)*sqrt(2*c*d - b*e)), 2/3*(sqrt(-c*e^2*x^2 - b
*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*((5*c^2*d*e - 4*b*c*e^2)*f - (c^2*d^2
- b^2*e^2)*g - 3*(c^2*e^2*f - c^2*d*e*g)*x)*sqrt(e*x + d) + 3*((c^3*e^4*f - c^3
*d*e^3*g)*x^3 - ((c^3*d*e^3 - 2*b*c^2*e^4)*f - (c^3*d^2*e^2 - 2*b*c^2*d*e^3)*g)*
x^2 + (c^3*d^3*e - 2*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (c^3*d^4 - 2*b*c^2*d^3*e +
b^2*c*d^2*e^2)*g - ((c^3*d^2*e^2 - b^2*c*e^4)*f - (c^3*d^3*e - b^2*c*d*e^3)*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^5*d^5*e^2 - 12*b*c^4*d^4*e^3
+ 13*b^2*c^3*d^3*e^4 - 6*b^3*c^2*d^2*e^5 + b^4*c*d*e^6 + (4*c^5*d^2*e^5 - 4*b*c
^4*d*e^6 + b^2*c^3*e^7)*x^3 - (4*c^5*d^3*e^4 - 12*b*c^4*d^2*e^5 + 9*b^2*c^3*d*e^
6 - 2*b^3*c^2*e^7)*x^2 - (4*c^5*d^4*e^3 - 4*b*c^4*d^3*e^4 - 3*b^2*c^3*d^2*e^5 +
4*b^3*c^2*d*e^6 - b^4*c*e^7)*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((e*x+d)**(3/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.695649, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="giac")

[Out]

sage0*x