3.23.72 \(\int \frac {\sqrt {d+e x} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\) [2272]

Optimal. Leaf size=155 \[ \frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{3/2}} \]

[Out]

-2*(-d*g+e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^(1/2)/(e*x+d)^(1/2))/e^2/(-b*e+2*c*d
)^(3/2)+2*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(1/2)/c/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {802, 674, 214} \begin {gather*} \frac {2 \sqrt {d+e x} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \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)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 802

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Dist[e*((m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(e f-d g) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(2 (e f-d g)) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )}{2 c d-b e}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{c e^2 (2 c d-b e) \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 {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 138, normalized size = 0.89 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (\sqrt {-2 c d+b e} (c e f+c d g-b e g)+c (e f-d g) \sqrt {-b e+c (d-e x)} \tan ^{-1}\left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )\right )}{c e^2 (-2 c d+b e)^{3/2} \sqrt {(d+e x) (-b e+c (d-e x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 199, normalized size = 1.28

method	result	size
default	\(-\frac {2 \left (\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d g \sqrt {-c e x -b e +c d}-\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c e f \sqrt {-c e x -b e +c d}+\sqrt {b e -2 c d}\, b e g -\sqrt {b e -2 c d}\, c d g -\sqrt {b e -2 c d}\, c e f \right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{\left (b e -2 c d \right )^{\frac {3}{2}} e^{2} c \left (c e x +b e -c d \right ) \sqrt {e x +d}}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2*(arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c*d*g*(-c*e*x-b*e+c*d)^(1/2)-arctan((-c*e*x-b*e+c*d)^(1/2
)/(b*e-2*c*d)^(1/2))*c*e*f*(-c*e*x-b*e+c*d)^(1/2)+(b*e-2*c*d)^(1/2)*b*e*g-(b*e-2*c*d)^(1/2)*c*d*g-(b*e-2*c*d)^
(1/2)*c*e*f)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(b*e-2*c*d)^(3/2)/e^2/c/(c*e*x+b*e-c*d)/(e*x+d)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (147) = 294\).
time = 2.42, size = 710, normalized size = 4.58 \begin {gather*} \left [\frac {{\left (c^{2} d^{3} g + {\left (c^{2} f x^{2} + b c f x\right )} e^{3} - {\left (c^{2} d g x^{2} + b c d g x - b c d f\right )} e^{2} - {\left (c^{2} d^{2} f + b c d^{2} g\right )} e\right )} \sqrt {2 \, c d - b e} \log \left (\frac {3 \, c d^{2} - {\left (c x^{2} + 2 \, b x\right )} e^{2} + 2 \, {\left (c d x - b d\right )} e + 2 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {2 \, c d - b e} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (2 \, c^{2} d^{2} g - {\left (b c f - b^{2} g\right )} e^{2} + {\left (2 \, c^{2} d f - 3 \, b c d g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{4 \, c^{4} d^{4} e^{2} - 8 \, b c^{3} d^{3} e^{3} - {\left (b^{2} c^{2} x^{2} + b^{3} c x\right )} e^{6} + {\left (4 \, b c^{3} d x^{2} + 4 \, b^{2} c^{2} d x - b^{3} c d\right )} e^{5} - {\left (4 \, c^{4} d^{2} x^{2} + 4 \, b c^{3} d^{2} x - 5 \, b^{2} c^{2} d^{2}\right )} e^{4}}, \frac {2 \, {\left ({\left (c^{2} d^{3} g + {\left (c^{2} f x^{2} + b c f x\right )} e^{3} - {\left (c^{2} d g x^{2} + b c d g x - b c d f\right )} e^{2} - {\left (c^{2} d^{2} f + b c d^{2} g\right )} e\right )} \sqrt {-2 \, c d + b e} \arctan \left (-\frac {\sqrt {-2 \, c d + b e} \sqrt {x e + d}}{\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}\right ) + {\left (2 \, c^{2} d^{2} g - {\left (b c f - b^{2} g\right )} e^{2} + {\left (2 \, c^{2} d f - 3 \, b c d g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}\right )}}{4 \, c^{4} d^{4} e^{2} - 8 \, b c^{3} d^{3} e^{3} - {\left (b^{2} c^{2} x^{2} + b^{3} c x\right )} e^{6} + {\left (4 \, b c^{3} d x^{2} + 4 \, b^{2} c^{2} d x - b^{3} c d\right )} e^{5} - {\left (4 \, c^{4} d^{2} x^{2} + 4 \, b c^{3} d^{2} x - 5 \, b^{2} c^{2} d^{2}\right )} e^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d + e x} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (147) = 294\).
time = 2.17, size = 326, normalized size = 2.10 \begin {gather*} -\frac {2 \, {\left (d g - f e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d e^{2} - b e^{3}\right )} \sqrt {-2 \, c d + b e}} + \frac {2 \, {\left (\sqrt {2 \, c d - b e} c d g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - \sqrt {2 \, c d - b e} c f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) e - \sqrt {-2 \, c d + b e} c d g - \sqrt {-2 \, c d + b e} c f e + \sqrt {-2 \, c d + b e} b g e\right )}}{2 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{2} d e^{2} - \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c e^{3}} + \frac {2 \, {\left (c d g + c f e - b g e\right )}}{{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(d + e*x)^(1/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),x)

[Out]

int(((f + g*x)*(d + e*x)^(1/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2), x)

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