3.23.82 \(\int \frac {\sqrt {d+e x} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [2282]

Optimal. Leaf size=313 \[ \frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 c e f-c d g-2 b e g}{3 c e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f-c d g-2 b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(5 c e f-c d g-2 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{7/2}} \]

[Out]

-(-2*b*e*g-c*d*g+5*c*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)^(7/2)+2/3*(-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)
^(3/2)+1/3*(2*b*e*g+c*d*g-5*c*e*f)/c/e^2/(-b*e+2*c*d)^2/(e*x+d)^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+(
-2*b*e*g-c*d*g+5*c*e*f)*(e*x+d)^(1/2)/e^2/(-b*e+2*c*d)^3/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)

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Rubi [A]
time = 0.27, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {802, 686, 680, 674, 214} \begin {gather*} \frac {\sqrt {d+e x} (-2 b e g-c d g+5 c e f)}{e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {-2 b e g-c d g+5 c e f}{3 c e^2 \sqrt {d+e x} (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 \sqrt {d+e x} (-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 b e g-c d g+5 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 )}{e^2 (2 c d-b e)^{7/2}} \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)^(5/2),x]

[Out]

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

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 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*c*d - b*e)*(d + e
*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(
b^2 - 4*a*c))), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), 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] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 686

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

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 )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{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 {(5 c e f-c d g-2 b e g) \int \frac {1}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{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 {5 c e f-c d g-2 b e g}{3 c e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f-c d g-2 b e g) \int \frac {\sqrt {d+e x}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{2 e (2 c d-b e)^2}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{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 {5 c e f-c d g-2 b e g}{3 c e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f-c d g-2 b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f-c d g-2 b e 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}{2 e (2 c d-b e)^3}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{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 {5 c e f-c d g-2 b e g}{3 c e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f-c d g-2 b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f-c d g-2 b e 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)^3}\\ &=\frac {2 (c e f+c d g-b e g) \sqrt {d+e x}}{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 {5 c e f-c d g-2 b e g}{3 c e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c e f-c d g-2 b e g) \sqrt {d+e x}}{e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(5 c e f-c d g-2 b e 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)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.88, size = 248, normalized size = 0.79 \begin {gather*} \frac {(d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (b^2 e^2 (-3 e f+11 d g+8 e g x)-2 b c e \left (4 d e f+9 d^2 g+e^2 x (10 f-3 g x)\right )+c^2 \left (7 d^3 g-15 e^3 f x^2+d^2 e (13 f-2 g x)+d e^2 x (10 f+3 g x)\right )\right )}{(2 c d-b e)^3 (d+e x)}+\frac {3 (-5 c e f+c d g+2 b e g) (-b e+c (d-e x))^{5/2} \tan ^{-1}\left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{7/2}}\right )}{3 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \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)^(5/2),x]

[Out]

((d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(b^2*e^2*(-3*e*f + 11*d*g + 8*e*g*x) - 2*b*c*e*(4*d*e*f + 9*d^2*g +
e^2*x*(10*f - 3*g*x)) + c^2*(7*d^3*g - 15*e^3*f*x^2 + d^2*e*(13*f - 2*g*x) + d*e^2*x*(10*f + 3*g*x))))/((2*c*d
- b*e)^3*(d + e*x)) + (3*(-5*c*e*f + c*d*g + 2*b*e*g)*(-(b*e) + c*(d - e*x))^(5/2)*ArcTan[Sqrt[-(b*e) + c*(d
- e*x)]/Sqrt[-2*c*d + b*e]])/(-2*c*d + b*e)^(7/2)))/(3*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(895\) vs. \(2(288)=576\).
time = 0.05, size = 896, normalized size = 2.86

method	result	size
default	\(-\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,e^{3} g \,x^{2}+3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} e^{3} f \,x^{2}+6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} e^{3} g x \sqrt {-c e x -b e +c d}+3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x -15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} f x \sqrt {-c e x -b e +c d}+6 \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}+3 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}+6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} d \,e^{2} g \sqrt {-c e x -b e +c d}-3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g -15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} f \sqrt {-c e x -b e +c d}-3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g +15 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f +8 \sqrt {b e -2 c d}\, b^{2} e^{3} g x -20 \sqrt {b e -2 c d}\, b c \,e^{3} f x -2 \sqrt {b e -2 c d}\, c^{2} d^{2} e g x +10 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x +11 \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g -3 \sqrt {b e -2 c d}\, b^{2} e^{3} f -18 \sqrt {b e -2 c d}\, b c \,d^{2} e g -8 \sqrt {b e -2 c d}\, b c d \,e^{2} f +7 \sqrt {b e -2 c d}\, c^{2} d^{3} g +13 \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{3 \left (e x +d \right )^{\frac {3}{2}} \left (c e x +b e -c d \right )^{2} e^{2} \left (b e -2 c d \right )^{\frac {7}{2}}}\)	\(896\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/
2)*b*c*e^3*g*x^2+3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*d*e^2*g*x^2-15*
arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*e^3*f*x^2+6*arctan((-c*e*x-b*e+c*d
)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*e^3*g*x*(-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*e*x-b*e+c*d)^(1/2)*b*c*d*e^2*g*x-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*e^3*f*x*(-c*e*
x-b*e+c*d)^(1/2)+6*(b*e-2*c*d)^(1/2)*b*c*e^3*g*x^2+3*(b*e-2*c*d)^(1/2)*c^2*d*e^2*g*x^2-15*(b*e-2*c*d)^(1/2)*c^
2*e^3*f*x^2+6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*d*e^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*e*x-b*e+c*d)^(1/2)*b*c*d^2*e*g-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*b*c*d*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*e*x-
b*e+c*d)^(1/2)*c^2*d^3*g+15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^2*d^2*e*
f+8*(b*e-2*c*d)^(1/2)*b^2*e^3*g*x-20*(b*e-2*c*d)^(1/2)*b*c*e^3*f*x-2*(b*e-2*c*d)^(1/2)*c^2*d^2*e*g*x+10*(b*e-2
*c*d)^(1/2)*c^2*d*e^2*f*x+11*(b*e-2*c*d)^(1/2)*b^2*d*e^2*g-3*(b*e-2*c*d)^(1/2)*b^2*e^3*f-18*(b*e-2*c*d)^(1/2)*
b*c*d^2*e*g-8*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f+7*(b*e-2*c*d)^(1/2)*c^2*d^3*g+13*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f)/(e
*x+d)^(3/2)/(c*e*x+b*e-c*d)^2/e^2/(b*e-2*c*d)^(7/2)

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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)^(5/2),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)^(5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (295) = 590\).
time = 3.10, size = 1951, normalized size = 6.23 \begin {gather*} \text {Too large to display} \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)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(c^3*d^5*g - 5*c^3*d^4*f*e - ((5*c^3*f - 2*b*c^2*g)*x^4 + 2*(5*b*c^2*f - 2*b^2*c*g)*x^3 + (5*b^2*c*f
- 2*b^3*g)*x^2)*e^5 + (c^3*d*g*x^4 + 2*b*c^2*d*g*x^3 - 5*(2*b*c^2*d*f - b^2*c*d*g)*x^2 - 2*(5*b^2*c*d*f - 2*b^
3*d*g)*x)*e^4 - (5*b^2*c*d^2*f - 2*b^3*d^2*g - 2*(5*c^3*d^2*f - b*c^2*d^2*g)*x^2 - 2*(5*b*c^2*d^2*f - b^2*c*d^
2*g)*x)*e^3 - (2*c^3*d^3*g*x^2 + 2*b*c^2*d^3*g*x - 10*b*c^2*d^3*f + 3*b^2*c*d^3*g)*e^2)*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*(14*c^3*d^4*g + (3*b^3*f + 3*(5*b*c^2*f - 2*b^2*c*g)*x^2 + 4*
(5*b^2*c*f - 2*b^3*g)*x)*e^4 + (2*b^2*c*d*f - 11*b^3*d*g - 3*(10*c^3*d*f - 3*b*c^2*d*g)*x^2 - 2*(25*b*c^2*d*f
- 8*b^2*c*d*g)*x)*e^3 + (6*c^3*d^2*g*x^2 - 29*b*c^2*d^2*f + 40*b^2*c*d^2*g + 2*(10*c^3*d^2*f + b*c^2*d^2*g)*x)
*e^2 - (4*c^3*d^3*g*x - 26*c^3*d^3*f + 43*b*c^2*d^3*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d
))/(16*c^6*d^8*e^2 - 64*b*c^5*d^7*e^3 + (b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*e^10 - 2*(4*b^3*c^3*d*x^4 + 8*b^
4*c^2*d*x^3 + 3*b^5*c*d*x^2 - b^6*d*x)*e^9 + (24*b^2*c^4*d^2*x^4 + 48*b^3*c^3*d^2*x^3 + 6*b^4*c^2*d^2*x^2 - 18
*b^5*c*d^2*x + b^6*d^2)*e^8 - 2*(16*b*c^5*d^3*x^4 + 32*b^2*c^4*d^3*x^3 - 16*b^3*c^3*d^3*x^2 - 32*b^4*c^2*d^3*x
+ 5*b^5*c*d^3)*e^7 + (16*c^6*d^4*x^4 + 32*b*c^5*d^4*x^3 - 96*b^2*c^4*d^4*x^2 - 112*b^3*c^3*d^4*x + 41*b^4*c^2
*d^4)*e^6 + 8*(12*b*c^5*d^5*x^2 + 12*b^2*c^4*d^5*x - 11*b^3*c^3*d^5)*e^5 - 8*(4*c^6*d^6*x^2 + 4*b*c^5*d^6*x -
13*b^2*c^4*d^6)*e^4), 1/3*(3*(c^3*d^5*g - 5*c^3*d^4*f*e - ((5*c^3*f - 2*b*c^2*g)*x^4 + 2*(5*b*c^2*f - 2*b^2*c*
g)*x^3 + (5*b^2*c*f - 2*b^3*g)*x^2)*e^5 + (c^3*d*g*x^4 + 2*b*c^2*d*g*x^3 - 5*(2*b*c^2*d*f - b^2*c*d*g)*x^2 - 2
*(5*b^2*c*d*f - 2*b^3*d*g)*x)*e^4 - (5*b^2*c*d^2*f - 2*b^3*d^2*g - 2*(5*c^3*d^2*f - b*c^2*d^2*g)*x^2 - 2*(5*b*
c^2*d^2*f - b^2*c*d^2*g)*x)*e^3 - (2*c^3*d^3*g*x^2 + 2*b*c^2*d^3*g*x - 10*b*c^2*d^3*f + 3*b^2*c*d^3*g)*e^2)*sq
rt(-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)) + (14*c^3*d
^4*g + (3*b^3*f + 3*(5*b*c^2*f - 2*b^2*c*g)*x^2 + 4*(5*b^2*c*f - 2*b^3*g)*x)*e^4 + (2*b^2*c*d*f - 11*b^3*d*g -
3*(10*c^3*d*f - 3*b*c^2*d*g)*x^2 - 2*(25*b*c^2*d*f - 8*b^2*c*d*g)*x)*e^3 + (6*c^3*d^2*g*x^2 - 29*b*c^2*d^2*f
+ 40*b^2*c*d^2*g + 2*(10*c^3*d^2*f + b*c^2*d^2*g)*x)*e^2 - (4*c^3*d^3*g*x - 26*c^3*d^3*f + 43*b*c^2*d^3*g)*e)*
sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))/(16*c^6*d^8*e^2 - 64*b*c^5*d^7*e^3 + (b^4*c^2*x^4 + 2*b
^5*c*x^3 + b^6*x^2)*e^10 - 2*(4*b^3*c^3*d*x^4 + 8*b^4*c^2*d*x^3 + 3*b^5*c*d*x^2 - b^6*d*x)*e^9 + (24*b^2*c^4*d
^2*x^4 + 48*b^3*c^3*d^2*x^3 + 6*b^4*c^2*d^2*x^2 - 18*b^5*c*d^2*x + b^6*d^2)*e^8 - 2*(16*b*c^5*d^3*x^4 + 32*b^2
*c^4*d^3*x^3 - 16*b^3*c^3*d^3*x^2 - 32*b^4*c^2*d^3*x + 5*b^5*c*d^3)*e^7 + (16*c^6*d^4*x^4 + 32*b*c^5*d^4*x^3 -
96*b^2*c^4*d^4*x^2 - 112*b^3*c^3*d^4*x + 41*b^4*c^2*d^4)*e^6 + 8*(12*b*c^5*d^5*x^2 + 12*b^2*c^4*d^5*x - 11*b^
3*c^3*d^5)*e^5 - 8*(4*c^6*d^6*x^2 + 4*b*c^5*d^6*x - 13*b^2*c^4*d^6)*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 {5}{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)**(5/2),x)

[Out]

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

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Giac [A]
time = 1.75, size = 379, normalized size = 1.21 \begin {gather*} -\frac {{\left (c d g - 5 \, c f e + 2 \, b g 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 (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (2 \, c^{2} d^{2} g + 2 \, c^{2} d f e - 3 \, b c d g e - b c f e^{2} + b^{2} g e^{2} - 6 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c f e + 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b g e\right )}}{3 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}} + \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c d g - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c f e}{{\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} {\left (x e + d\right )} c} \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)^(5/2),x, algorithm="giac")

[Out]

-(c*d*g - 5*c*f*e + 2*b*g*e)*arctan(sqrt(-(x*e + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((8*c^3*d^3*e^2 - 12*
b*c^2*d^2*e^3 + 6*b^2*c*d*e^4 - b^3*e^5)*sqrt(-2*c*d + b*e)) - 2/3*(2*c^2*d^2*g + 2*c^2*d*f*e - 3*b*c*d*g*e -
b*c*f*e^2 + b^2*g*e^2 - 6*((x*e + d)*c - 2*c*d + b*e)*c*f*e + 3*((x*e + d)*c - 2*c*d + b*e)*b*g*e)/((8*c^3*d^3
*e^2 - 12*b*c^2*d^2*e^3 + 6*b^2*c*d*e^4 - b^3*e^5)*((x*e + d)*c - 2*c*d + b*e)*sqrt(-(x*e + d)*c + 2*c*d - b*e
)) + (sqrt(-(x*e + d)*c + 2*c*d - b*e)*c*d*g - sqrt(-(x*e + d)*c + 2*c*d - b*e)*c*f*e)/((8*c^3*d^3*e^2 - 12*b*
c^2*d^2*e^3 + 6*b^2*c*d*e^4 - b^3*e^5)*(x*e + d)*c)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \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 )}^{5/2}} \,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)^(5/2),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)^(5/2), x)

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