3.23.83 \(\int \frac {f+g x}{\sqrt {d+e x} (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [2283]

Optimal. Leaf size=378 \[ -\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 (7 c e f+c d g-4 b e g)}{12 e^2 (2 c d-b e)^3 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {5 c (7 c e f+c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c (7 c e f+c d g-4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.39, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {806, 680, 686, 674, 214} \begin {gather*} -\frac {e f-d g}{2 e^2 \sqrt {d+e x} (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 \sqrt {d+e x} (-4 b e g+c d g+7 c e f)}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 (-4 b e g+c d g+7 c e f)}{12 e^2 \sqrt {d+e x} (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {\sqrt {d+e x} (-4 b e g+c d g+7 c e f)}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 c (-4 b e g+c d g+7 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)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 806

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

Rubi steps

\begin {align*} \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 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 )^{5/2}} \, dx}{4 e (2 c d-b e)}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(5 (7 c e f+c d g-4 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}{12 e (2 c d-b e)^2}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 (7 c e f+c d g-4 b e g)}{12 e^2 (2 c d-b e)^3 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c (7 c e f+c d g-4 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}{8 e (2 c d-b e)^3}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 (7 c e f+c d g-4 b e g)}{12 e^2 (2 c d-b e)^3 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {5 c (7 c e f+c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c (7 c e f+c d g-4 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}{8 e (2 c d-b e)^4}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 (7 c e f+c d g-4 b e g)}{12 e^2 (2 c d-b e)^3 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {5 c (7 c e f+c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(5 c (7 c e f+c d g-4 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 )}{4 (2 c d-b e)^4}\\ &=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 (7 c e f+c d g-4 b e g)}{12 e^2 (2 c d-b e)^3 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {5 c (7 c e f+c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c (7 c e f+c d g-4 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 )}{4 e^2 (2 c d-b e)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.24, size = 329, normalized size = 0.87 \begin {gather*} \frac {c (d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (6 b^3 e^3 (d g+e (f+2 g x))+c^3 \left (61 d^4 g-105 e^4 f x^3+d^2 e^2 x (161 f-5 g x)-5 d e^3 x^2 (7 f+3 g x)+d^3 e (43 f+23 g x)\right )-4 b c^2 e \left (33 d^3 g+49 d e^2 f x+5 e^3 x^2 (7 f-3 g x)+d^2 e (-4 f+30 g x)\right )+b^2 c e^2 \left (65 d^2 g+e^2 x (-21 f+80 g x)+d e (-57 f+109 g x)\right )\right )}{c (-2 c d+b e)^4 (d+e x)^2}+\frac {15 (7 c e f+c d g-4 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)^{9/2}}\right )}{12 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(6*b^3*e^3*(d*g + e*(f + 2*g*x)) + c^3*(61*d^4*g - 105*e^4*f*x^3 +
d^2*e^2*x*(161*f - 5*g*x) - 5*d*e^3*x^2*(7*f + 3*g*x) + d^3*e*(43*f + 23*g*x)) - 4*b*c^2*e*(33*d^3*g + 49*d*e
^2*f*x + 5*e^3*x^2*(7*f - 3*g*x) + d^2*e*(-4*f + 30*g*x)) + b^2*c*e^2*(65*d^2*g + e^2*x*(-21*f + 80*g*x) + d*e
*(-57*f + 109*g*x))))/(c*(-2*c*d + b*e)^4*(d + e*x)^2) + (15*(7*c*e*f + c*d*g - 4*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)^(9/2)))/(12*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. \(1519\) vs. \(2(344)=688\).
time = 0.05, size = 1520, normalized size = 4.02

method	result	size
default	\(\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (-105 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,c^{2} d^{2} e^{2} f \sqrt {-c e x -b e +c d}-35 \sqrt {b e -2 c d}\, c^{3} d \,e^{3} f \,x^{2}-21 \sqrt {b e -2 c d}\, b^{2} c \,e^{4} f x +23 \sqrt {b e -2 c d}\, c^{3} d^{3} e g x +161 \sqrt {b e -2 c d}\, c^{3} d^{2} e^{2} f x +105 \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^{3} d^{3} e f +65 \sqrt {b e -2 c d}\, b^{2} c \,d^{2} e^{2} g -57 \sqrt {b e -2 c d}\, b^{2} c d \,e^{3} f -132 \sqrt {b e -2 c d}\, b \,c^{2} d^{3} e g +16 \sqrt {b e -2 c d}\, b \,c^{2} d^{2} e^{2} f -105 \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^{3} e^{4} f \,x^{3}+60 \sqrt {b e -2 c d}\, b \,c^{2} e^{4} g \,x^{3}-15 \sqrt {b e -2 c d}\, c^{3} d \,e^{3} g \,x^{3}+80 \sqrt {b e -2 c d}\, b^{2} c \,e^{4} g \,x^{2}-140 \sqrt {b e -2 c d}\, b \,c^{2} e^{4} f \,x^{2}-5 \sqrt {b e -2 c d}\, c^{3} d^{2} e^{2} g \,x^{2}+6 \sqrt {b e -2 c d}\, b^{3} e^{4} f +61 \sqrt {b e -2 c d}\, c^{3} d^{4} g +60 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} c \,e^{4} g \,x^{2} \sqrt {-c e x -b e +c d}-105 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,c^{2} e^{4} f \,x^{2} \sqrt {-c e x -b e +c d}+60 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} c \,d^{2} e^{2} g \sqrt {-c e x -b e +c d}+6 \sqrt {b e -2 c d}\, b^{3} d \,e^{3} g +43 \sqrt {b e -2 c d}\, c^{3} d^{3} e f -105 \sqrt {b e -2 c d}\, c^{3} e^{4} f \,x^{3}+12 \sqrt {b e -2 c d}\, b^{3} e^{4} g x +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^{3} d^{4} g +45 \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^{2} d \,e^{3} g \,x^{2}-90 \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^{2} d^{2} e^{2} g x -105 \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^{3} d \,e^{3} f \,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^{3} d^{3} e g x +105 \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^{3} d^{2} e^{2} f x +109 \sqrt {b e -2 c d}\, b^{2} c d \,e^{3} g x -120 \sqrt {b e -2 c d}\, b \,c^{2} d^{2} e^{2} g x -196 \sqrt {b e -2 c d}\, b \,c^{2} d \,e^{3} f x -75 \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^{2} d^{3} e g +120 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} c d \,e^{3} g x \sqrt {-c e x -b e +c d}-210 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,c^{2} d \,e^{3} f x \sqrt {-c e x -b e +c d}+60 \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^{2} e^{4} g \,x^{3}-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^{3} d \,e^{3} g \,x^{3}-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^{3} d^{2} e^{2} g \,x^{2}\right )}{12 \left (e x +d \right )^{\frac {5}{2}} \left (c e x +b e -c d \right )^{2} e^{2} \left (b e -2 c d \right )^{\frac {9}{2}}}\)	\(1520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/12*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*f*(
-c*e*x-b*e+c*d)^(1/2)-35*(b*e-2*c*d)^(1/2)*c^3*d*e^3*f*x^2-21*(b*e-2*c*d)^(1/2)*b^2*c*e^4*f*x+23*(b*e-2*c*d)^(
1/2)*c^3*d^3*e*g*x+161*(b*e-2*c*d)^(1/2)*c^3*d^2*e^2*f*x+105*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^3*d^3*e*f+65*(b*e-2*c*d)^(1/2)*b^2*c*d^2*e^2*g-57*(b*e-2*c*d)^(1/2)*b^2*c*d*e^3*f-132
*(b*e-2*c*d)^(1/2)*b*c^2*d^3*e*g+16*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*f-105*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^3*e^4*f*x^3+60*(b*e-2*c*d)^(1/2)*b*c^2*e^4*g*x^3-15*(b*e-2*c*d)^(1/2)*c^
3*d*e^3*g*x^3+80*(b*e-2*c*d)^(1/2)*b^2*c*e^4*g*x^2-140*(b*e-2*c*d)^(1/2)*b*c^2*e^4*f*x^2-5*(b*e-2*c*d)^(1/2)*c
^3*d^2*e^2*g*x^2+6*(b*e-2*c*d)^(1/2)*b^3*e^4*f+61*(b*e-2*c*d)^(1/2)*c^3*d^4*g+60*arctan((-c*e*x-b*e+c*d)^(1/2)
/(b*e-2*c*d)^(1/2))*b^2*c*e^4*g*x^2*(-c*e*x-b*e+c*d)^(1/2)-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*b*c^2*e^4*f*x^2*(-c*e*x-b*e+c*d)^(1/2)+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d^2*e^2*g*(
-c*e*x-b*e+c*d)^(1/2)+6*(b*e-2*c*d)^(1/2)*b^3*d*e^3*g+43*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f-105*(b*e-2*c*d)^(1/2)*c
^3*e^4*f*x^3+12*(b*e-2*c*d)^(1/2)*b^3*e^4*g*x+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^3*d^4*g+45*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^2*d*e^3*g*
x^2-90*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^2*d^2*e^2*g*x-105*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^3*d*e^3*f*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^3*d^3*e*g*x+105*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^3*d^2*e^2*f*x+109*(b*e-2*c*d)^(1/2)*b^2*c*d*e^3*g*x-120*(b*e-2*c*d)^(1/2)*b*c^2*d^2*
e^2*g*x-196*(b*e-2*c*d)^(1/2)*b*c^2*d*e^3*f*x-75*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^2*d^3*e*g+120*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3*g*x*(-c*e*x-b*e+c*d)
^(1/2)-210*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d*e^3*f*x*(-c*e*x-b*e+c*d)^(1/2)+60*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^2*e^4*g*x^3-15*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^3*d*e^3*g*x^3-15*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^3*d^2*e^2*g*x^2)/(e*x+d)^(5/2)/(c*e*x+b*e-c*d)^2/e^2/(b*e-2*c*d)^(9/2)

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1442 vs. \(2 (354) = 708\).
time = 3.65, size = 2940, normalized size = 7.78 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/24*(15*(c^4*d^6*g + ((7*c^4*f - 4*b*c^3*g)*x^5 + 2*(7*b*c^3*f - 4*b^2*c^2*g)*x^4 + (7*b^2*c^2*f - 4*b^3*c*
g)*x^3)*e^6 + (c^4*d*g*x^5 + (7*c^4*d*f - 2*b*c^3*d*g)*x^4 + (28*b*c^3*d*f - 15*b^2*c^2*d*g)*x^3 + 3*(7*b^2*c^
2*d*f - 4*b^3*c*d*g)*x^2)*e^5 + (c^4*d^2*g*x^4 + 3*b^2*c^2*d^2*g*x^2 - 2*(7*c^4*d^2*f - 6*b*c^3*d^2*g)*x^3 + 3
*(7*b^2*c^2*d^2*f - 4*b^3*c*d^2*g)*x)*e^4 - (2*c^4*d^3*g*x^3 - 7*b^2*c^2*d^3*f + 4*b^3*c*d^3*g + 2*(7*c^4*d^3*
f - 4*b*c^3*d^3*g)*x^2 + (28*b*c^3*d^3*f - 19*b^2*c^2*d^3*g)*x)*e^3 - (2*c^4*d^4*g*x^2 + 14*b*c^3*d^4*f - 9*b^
2*c^2*d^4*g - (7*c^4*d^4*f - 8*b*c^3*d^4*g)*x)*e^2 + (c^4*d^5*g*x + 7*c^4*d^5*f - 6*b*c^3*d^5*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)*sqr
t(2*c*d - b*e)*sqrt(x*e + d))/(x^2*e^2 + 2*d*x*e + d^2)) - 2*(122*c^4*d^5*g - (6*b^4*f - 15*(7*b*c^3*f - 4*b^2
*c^2*g)*x^3 - 20*(7*b^2*c^2*f - 4*b^3*c*g)*x^2 - 3*(7*b^3*c*f - 4*b^4*g)*x)*e^5 + (69*b^3*c*d*f - 6*b^4*d*g -
15*(14*c^4*d*f - 9*b*c^3*d*g)*x^3 - 5*(49*b*c^3*d*f - 32*b^2*c^2*d*g)*x^2 + (154*b^2*c^2*d*f - 85*b^3*c*d*g)*x
)*e^4 - (30*c^4*d^2*g*x^3 + 130*b^2*c^2*d^2*f + 53*b^3*c*d^2*g + 5*(14*c^4*d^2*f - b*c^3*d^2*g)*x^2 + (553*b*c
^3*d^2*f - 338*b^2*c^2*d^2*g)*x)*e^3 - (10*c^4*d^3*g*x^2 + 11*b*c^3*d^3*f - 262*b^2*c^2*d^3*g - (322*c^4*d^3*f
- 263*b*c^3*d^3*g)*x)*e^2 + (46*c^4*d^4*g*x + 86*c^4*d^4*f - 325*b*c^3*d^4*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2
+ b*x)*e^2)*sqrt(x*e + d))/(32*c^7*d^10*e^2 - (b^5*c^2*x^5 + 2*b^6*c*x^4 + b^7*x^3)*e^12 + (10*b^4*c^3*d*x^5 +
19*b^5*c^2*d*x^4 + 6*b^6*c*d*x^3 - 3*b^7*d*x^2)*e^11 - (40*b^3*c^4*d^2*x^5 + 70*b^4*c^3*d^2*x^4 - 2*b^5*c^2*d
^2*x^3 - 30*b^6*c*d^2*x^2 + 3*b^7*d^2*x)*e^10 + (80*b^2*c^5*d^3*x^5 + 120*b^3*c^4*d^3*x^4 - 100*b^4*c^3*d^3*x^
3 - 118*b^5*c^2*d^3*x^2 + 34*b^6*c*d^3*x - b^7*d^3)*e^9 - (80*b*c^6*d^4*x^5 + 80*b^2*c^5*d^4*x^4 - 320*b^3*c^4
*d^4*x^3 - 220*b^4*c^3*d^4*x^2 + 161*b^5*c^2*d^4*x - 12*b^6*c*d^4)*e^8 + (32*c^7*d^5*x^5 - 16*b*c^6*d^5*x^4 -
448*b^2*c^5*d^5*x^3 - 160*b^3*c^4*d^5*x^2 + 410*b^4*c^3*d^5*x - 61*b^5*c^2*d^5)*e^7 + 2*(16*c^7*d^6*x^4 + 144*
b*c^6*d^6*x^3 - 32*b^2*c^5*d^6*x^2 - 300*b^3*c^4*d^6*x + 85*b^4*c^3*d^6)*e^6 - 8*(8*c^7*d^7*x^3 - 20*b*c^6*d^7
*x^2 - 62*b^2*c^5*d^7*x + 35*b^3*c^4*d^7)*e^5 - 16*(4*c^7*d^8*x^2 + 13*b*c^6*d^8*x - 17*b^2*c^5*d^8)*e^4 + 16*
(2*c^7*d^9*x - 9*b*c^6*d^9)*e^3), -1/12*(15*(c^4*d^6*g + ((7*c^4*f - 4*b*c^3*g)*x^5 + 2*(7*b*c^3*f - 4*b^2*c^2
*g)*x^4 + (7*b^2*c^2*f - 4*b^3*c*g)*x^3)*e^6 + (c^4*d*g*x^5 + (7*c^4*d*f - 2*b*c^3*d*g)*x^4 + (28*b*c^3*d*f -
15*b^2*c^2*d*g)*x^3 + 3*(7*b^2*c^2*d*f - 4*b^3*c*d*g)*x^2)*e^5 + (c^4*d^2*g*x^4 + 3*b^2*c^2*d^2*g*x^2 - 2*(7*c
^4*d^2*f - 6*b*c^3*d^2*g)*x^3 + 3*(7*b^2*c^2*d^2*f - 4*b^3*c*d^2*g)*x)*e^4 - (2*c^4*d^3*g*x^3 - 7*b^2*c^2*d^3*
f + 4*b^3*c*d^3*g + 2*(7*c^4*d^3*f - 4*b*c^3*d^3*g)*x^2 + (28*b*c^3*d^3*f - 19*b^2*c^2*d^3*g)*x)*e^3 - (2*c^4*
d^4*g*x^2 + 14*b*c^3*d^4*f - 9*b^2*c^2*d^4*g - (7*c^4*d^4*f - 8*b*c^3*d^4*g)*x)*e^2 + (c^4*d^5*g*x + 7*c^4*d^5
*f - 6*b*c^3*d^5*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)) - (122*c^4*d^5*g - (6*b^4*f - 15*(7*b*c^3*f - 4*b^2*c^2*g)*x^3 - 20*(7*b^2*c^2*f - 4*b^3*c*g)*x^
2 - 3*(7*b^3*c*f - 4*b^4*g)*x)*e^5 + (69*b^3*c*d*f - 6*b^4*d*g - 15*(14*c^4*d*f - 9*b*c^3*d*g)*x^3 - 5*(49*b*c
^3*d*f - 32*b^2*c^2*d*g)*x^2 + (154*b^2*c^2*d*f - 85*b^3*c*d*g)*x)*e^4 - (30*c^4*d^2*g*x^3 + 130*b^2*c^2*d^2*f
+ 53*b^3*c*d^2*g + 5*(14*c^4*d^2*f - b*c^3*d^2*g)*x^2 + (553*b*c^3*d^2*f - 338*b^2*c^2*d^2*g)*x)*e^3 - (10*c^
4*d^3*g*x^2 + 11*b*c^3*d^3*f - 262*b^2*c^2*d^3*g - (322*c^4*d^3*f - 263*b*c^3*d^3*g)*x)*e^2 + (46*c^4*d^4*g*x
+ 86*c^4*d^4*f - 325*b*c^3*d^4*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))/(32*c^7*d^10*e^2 -
(b^5*c^2*x^5 + 2*b^6*c*x^4 + b^7*x^3)*e^12 + (10*b^4*c^3*d*x^5 + 19*b^5*c^2*d*x^4 + 6*b^6*c*d*x^3 - 3*b^7*d*x
^2)*e^11 - (40*b^3*c^4*d^2*x^5 + 70*b^4*c^3*d^2*x^4 - 2*b^5*c^2*d^2*x^3 - 30*b^6*c*d^2*x^2 + 3*b^7*d^2*x)*e^10
+ (80*b^2*c^5*d^3*x^5 + 120*b^3*c^4*d^3*x^4 - 100*b^4*c^3*d^3*x^3 - 118*b^5*c^2*d^3*x^2 + 34*b^6*c*d^3*x - b^
7*d^3)*e^9 - (80*b*c^6*d^4*x^5 + 80*b^2*c^5*d^4*x^4 - 320*b^3*c^4*d^4*x^3 - 220*b^4*c^3*d^4*x^2 + 161*b^5*c^2*
d^4*x - 12*b^6*c*d^4)*e^8 + (32*c^7*d^5*x^5 - 16*b*c^6*d^5*x^4 - 448*b^2*c^5*d^5*x^3 - 160*b^3*c^4*d^5*x^2 + 4
10*b^4*c^3*d^5*x - 61*b^5*c^2*d^5)*e^7 + 2*(16*c^7*d^6*x^4 + 144*b*c^6*d^6*x^3 - 32*b^2*c^5*d^6*x^2 - 300*b^3*
c^4*d^6*x + 85*b^4*c^3*d^6)*e^6 - 8*(8*c^7*d^7*x^3 - 20*b*c^6*d^7*x^2 - 62*b^2*c^5*d^7*x + 35*b^3*c^4*d^7)*e^5
- 16*(4*c^7*d^8*x^2 + 13*b*c^6*d^8*x - 17*b^2*c^5*d^8)*e^4 + 16*(2*c^7*d^9*x - 9*b*c^6*d^9)*e^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)**(1/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

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

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Giac [A]
time = 2.38, size = 637, normalized size = 1.69 \begin {gather*} \frac {5 \, {\left (c^{2} d g + 7 \, c^{2} f e - 4 \, b c 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 )}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (2 \, c^{3} d^{2} g + 2 \, c^{3} d f e - 3 \, b c^{2} d g e - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c^{2} d g - b c^{2} f e^{2} + b^{2} c g e^{2} - 9 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} c^{2} f e + 6 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )} b c g e\right )}}{3 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\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 {10 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 26 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} d f e + 3 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} d g e - 3 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 13 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{2} f e^{2} - 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c g e^{2} + 11 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} f e - 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c g e}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} {\left (x e + d\right )}^{2} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

5/4*(c^2*d*g + 7*c^2*f*e - 4*b*c*g*e)*arctan(sqrt(-(x*e + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((16*c^4*d^4
*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*sqrt(-2*c*d + b*e)) - 2/3*(2*c^3*d^2*g
+ 2*c^3*d*f*e - 3*b*c^2*d*g*e - 3*((x*e + d)*c - 2*c*d + b*e)*c^2*d*g - b*c^2*f*e^2 + b^2*c*g*e^2 - 9*((x*e +
d)*c - 2*c*d + b*e)*c^2*f*e + 6*((x*e + d)*c - 2*c*d + b*e)*b*c*g*e)/((16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24
*b^2*c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*((x*e + d)*c - 2*c*d + b*e)*sqrt(-(x*e + d)*c + 2*c*d - b*e)) + 1/
4*(10*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^3*d^2*g - 26*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^3*d*f*e + 3*sqrt(-(x*
e + d)*c + 2*c*d - b*e)*b*c^2*d*g*e - 3*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^2*d*g + 13*sqrt(-(x*e + d)*c + 2*
c*d - b*e)*b*c^2*f*e^2 - 4*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b^2*c*g*e^2 + 11*(-(x*e + d)*c + 2*c*d - b*e)^(3/2
)*c^2*f*e - 4*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c*g*e)/((16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2
*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*(x*e + d)^2*c^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {f+g\,x}{\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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