3.23.37 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\) [2237]

Optimal. Leaf size=231 \[ -\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (c e f+7 c d g-4 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.24, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {806, 676, 674, 214} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {c (-4 b e g+7 c d g+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)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((c*e*f + 7*c*d*g - 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(2*c*d - b*e)*(d + e*x)^(3/2
)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) + (c*(c*e
*f + 7*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)^(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 676

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + p + 1))), x] - Dist[c*(p/(e^2*(m + p + 1))), Int[(d + e*x)^(m + 2)*(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] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 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 {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {(c e f+7 c d g-4 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx}{4 e (2 c d-b e)}\\ &=-\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(c (c e f+7 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)}\\ &=-\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(c (c e f+7 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)}\\ &=-\frac {(c e f+7 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (c e f+7 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)^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.50, size = 181, normalized size = 0.78 \begin {gather*} \frac {c \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {2 b e (d g+e (f+2 g x))+c \left (-5 d^2 g+e^2 f x-3 d e (f+3 g x)\right )}{c (2 c d-b e) (d+e x)^2}+\frac {(c e f+7 c d g-4 b e g) \tan ^{-1}\left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{3/2} \sqrt {-b e+c (d-e x)}}\right )}{4 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(621\) vs. \(2(209)=418\).
time = 0.04, size = 622, normalized size = 2.69

method	result	size
default	\(-\frac {\left (4 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}-7 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}-\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}+8 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -14 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -2 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x +4 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -7 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +4 b \,e^{2} g x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-9 c d e g x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+c \,e^{2} f x \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+2 b d e g \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+2 b \,e^{2} f \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-5 c \,d^{2} g \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}-3 c d e f \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{4 \left (b e -2 c d \right )^{\frac {3}{2}} e^{2} \sqrt {-c e x -b e +c d}\, \left (e x +d \right )^{\frac {5}{2}}}\)	\(622\)

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)^(1/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*(4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*e^3*g*x^2-7*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*
c*d)^(1/2))*c^2*d*e^2*g*x^2-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*e^3*f*x^2+8*arctan((-c*e*x-b*
e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e^2*g*x-14*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*
x-2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e^2*f*x+4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)
^(1/2))*b*c*d^2*e*g-7*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g-arctan((-c*e*x-b*e+c*d)^(1/2)
/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f+4*b*e^2*g*x*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-9*c*d*e*g*x*(b*e-2*c*d)^(
1/2)*(-c*e*x-b*e+c*d)^(1/2)+c*e^2*f*x*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+2*b*d*e*g*(b*e-2*c*d)^(1/2)*(-c
*e*x-b*e+c*d)^(1/2)+2*b*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-5*c*d^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e
+c*d)^(1/2)-3*c*d*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2))*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(b*e-2*c*d)^(
3/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)/(e*x+d)^(5/2)

________________________________________________________________________________________

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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(g*x + f)/(x*e + d)^(7/2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (215) = 430\).
time = 2.98, size = 964, normalized size = 4.17 \begin {gather*} \left [\frac {{\left (7 \, c^{2} d^{4} g + {\left (c^{2} f - 4 \, b c g\right )} x^{3} e^{4} + {\left (7 \, c^{2} d g x^{3} + 3 \, {\left (c^{2} d f - 4 \, b c d g\right )} x^{2}\right )} e^{3} + 3 \, {\left (7 \, c^{2} d^{2} g x^{2} + {\left (c^{2} d^{2} f - 4 \, b c d^{2} g\right )} x\right )} e^{2} + {\left (21 \, c^{2} d^{3} g x + c^{2} d^{3} f - 4 \, b c d^{3} 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 (10 \, c^{2} d^{3} g + {\left (2 \, b^{2} f + {\left (b c f + 4 \, b^{2} g\right )} x\right )} e^{3} - {\left (7 \, b c d f - 2 \, b^{2} d g + {\left (2 \, c^{2} d f + 17 \, b c d g\right )} x\right )} e^{2} + 3 \, {\left (6 \, c^{2} d^{2} g x + 2 \, c^{2} d^{2} f - 3 \, b c d^{2} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} \sqrt {x e + d}}{8 \, {\left (4 \, c^{2} d^{5} e^{2} + b^{2} x^{3} e^{7} - {\left (4 \, b c d x^{3} - 3 \, b^{2} d x^{2}\right )} e^{6} + {\left (4 \, c^{2} d^{2} x^{3} - 12 \, b c d^{2} x^{2} + 3 \, b^{2} d^{2} x\right )} e^{5} + {\left (12 \, c^{2} d^{3} x^{2} - 12 \, b c d^{3} x + b^{2} d^{3}\right )} e^{4} + 4 \, {\left (3 \, c^{2} d^{4} x - b c d^{4}\right )} e^{3}\right )}}, \frac {{\left (7 \, c^{2} d^{4} g + {\left (c^{2} f - 4 \, b c g\right )} x^{3} e^{4} + {\left (7 \, c^{2} d g x^{3} + 3 \, {\left (c^{2} d f - 4 \, b c d g\right )} x^{2}\right )} e^{3} + 3 \, {\left (7 \, c^{2} d^{2} g x^{2} + {\left (c^{2} d^{2} f - 4 \, b c d^{2} g\right )} x\right )} e^{2} + {\left (21 \, c^{2} d^{3} g x + c^{2} d^{3} f - 4 \, b c d^{3} 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 (10 \, c^{2} d^{3} g + {\left (2 \, b^{2} f + {\left (b c f + 4 \, b^{2} g\right )} x\right )} e^{3} - {\left (7 \, b c d f - 2 \, b^{2} d g + {\left (2 \, c^{2} d f + 17 \, b c d g\right )} x\right )} e^{2} + 3 \, {\left (6 \, c^{2} d^{2} g x + 2 \, c^{2} d^{2} f - 3 \, b c d^{2} 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 \, {\left (4 \, c^{2} d^{5} e^{2} + b^{2} x^{3} e^{7} - {\left (4 \, b c d x^{3} - 3 \, b^{2} d x^{2}\right )} e^{6} + {\left (4 \, c^{2} d^{2} x^{3} - 12 \, b c d^{2} x^{2} + 3 \, b^{2} d^{2} x\right )} e^{5} + {\left (12 \, c^{2} d^{3} x^{2} - 12 \, b c d^{3} x + b^{2} d^{3}\right )} e^{4} + 4 \, {\left (3 \, c^{2} d^{4} x - b c d^{4}\right )} e^{3}\right )}}\right ] \end {gather*}

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)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

[1/8*((7*c^2*d^4*g + (c^2*f - 4*b*c*g)*x^3*e^4 + (7*c^2*d*g*x^3 + 3*(c^2*d*f - 4*b*c*d*g)*x^2)*e^3 + 3*(7*c^2*
d^2*g*x^2 + (c^2*d^2*f - 4*b*c*d^2*g)*x)*e^2 + (21*c^2*d^3*g*x + c^2*d^3*f - 4*b*c*d^3*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*(10*c^2*d^3*g + (2*b^2*f + (b*c*f + 4*b^2*g)*x)*e^3 - (7
*b*c*d*f - 2*b^2*d*g + (2*c^2*d*f + 17*b*c*d*g)*x)*e^2 + 3*(6*c^2*d^2*g*x + 2*c^2*d^2*f - 3*b*c*d^2*g)*e)*sqrt
(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))/(4*c^2*d^5*e^2 + b^2*x^3*e^7 - (4*b*c*d*x^3 - 3*b^2*d*x^2)*
e^6 + (4*c^2*d^2*x^3 - 12*b*c*d^2*x^2 + 3*b^2*d^2*x)*e^5 + (12*c^2*d^3*x^2 - 12*b*c*d^3*x + b^2*d^3)*e^4 + 4*(
3*c^2*d^4*x - b*c*d^4)*e^3), 1/4*((7*c^2*d^4*g + (c^2*f - 4*b*c*g)*x^3*e^4 + (7*c^2*d*g*x^3 + 3*(c^2*d*f - 4*b
*c*d*g)*x^2)*e^3 + 3*(7*c^2*d^2*g*x^2 + (c^2*d^2*f - 4*b*c*d^2*g)*x)*e^2 + (21*c^2*d^3*g*x + c^2*d^3*f - 4*b*c
*d^3*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)
) - (10*c^2*d^3*g + (2*b^2*f + (b*c*f + 4*b^2*g)*x)*e^3 - (7*b*c*d*f - 2*b^2*d*g + (2*c^2*d*f + 17*b*c*d*g)*x)
*e^2 + 3*(6*c^2*d^2*g*x + 2*c^2*d^2*f - 3*b*c*d^2*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*sqrt(x*e + d))
/(4*c^2*d^5*e^2 + b^2*x^3*e^7 - (4*b*c*d*x^3 - 3*b^2*d*x^2)*e^6 + (4*c^2*d^2*x^3 - 12*b*c*d^2*x^2 + 3*b^2*d^2*
x)*e^5 + (12*c^2*d^3*x^2 - 12*b*c*d^3*x + b^2*d^3)*e^4 + 4*(3*c^2*d^4*x - b*c*d^4)*e^3)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}

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)**(1/2)/(e*x+d)**(7/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.87, size = 354, normalized size = 1.53 \begin {gather*} -\frac {{\left (\frac {{\left (7 \, c^{3} d g + c^{3} f e - 4 \, b c^{2} 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 (2 \, c d - b e\right )} \sqrt {-2 \, c d + b e}} + \frac {14 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d^{2} g + 2 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d f e - 15 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} d g e - 9 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} d g - \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} f e^{2} + 4 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{2} g e^{2} + {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} f e + 4 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{2} g e}{{\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{2} c^{2}}\right )} e^{\left (-2\right )}}{4 \, c} \end {gather*}

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)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

-1/4*((7*c^3*d*g + c^3*f*e - 4*b*c^2*g*e)*arctan(sqrt(-(x*e + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((2*c*d
- b*e)*sqrt(-2*c*d + b*e)) + (14*sqrt(-(x*e + d)*c + 2*c*d - b*e)*c^4*d^2*g + 2*sqrt(-(x*e + d)*c + 2*c*d - b*
e)*c^4*d*f*e - 15*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*c^3*d*g*e - 9*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*c^3*d*g
- sqrt(-(x*e + d)*c + 2*c*d - b*e)*b*c^3*f*e^2 + 4*sqrt(-(x*e + d)*c + 2*c*d - b*e)*b^2*c^2*g*e^2 + (-(x*e + d
)*c + 2*c*d - b*e)^(3/2)*c^3*f*e + 4*(-(x*e + d)*c + 2*c*d - b*e)^(3/2)*b*c^2*g*e)/((2*c*d - b*e)*(x*e + d)^2*
c^2))*e^(-2)/c

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________