3.2284 \(\int \frac{f+g x}{(d+e x)^{3/2} (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=457 \[ \frac{35 c^2 \sqrt{d+e x} (-2 b e g+c d g+3 c e f)}{8 e^2 (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{35 c^2 (-2 b e g+c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{11/2}}-\frac{35 c (-2 b e g+c d g+3 c e f)}{24 e^2 \sqrt{d+e x} (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{7 c \sqrt{d+e x} (-2 b e g+c d g+3 c e f)}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{-2 b e g+c d g+3 c e f}{4 e^2 \sqrt{d+e x} (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
[Out]
-(e*f - d*g)/(3*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (3*c*e*f + c*
d*g - 2*b*e*g)/(4*e^2*(2*c*d - b*e)^2*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (7*c*(3*c*e
*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x])/(12*e^2*(2*c*d - b*e)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (3
5*c*(3*c*e*f + c*d*g - 2*b*e*g))/(24*e^2*(2*c*d - b*e)^4*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2]) + (35*c^2*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x])/(8*e^2*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2]) - (35*c^2*(3*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])])/(8*e^2*(2*c*d - b*e)^(11/2))
________________________________________________________________________________________
Rubi [A] time = 0.752925, antiderivative size = 457, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109, Rules used =
{792, 672, 666, 660, 208} \[ \frac{35 c^2 \sqrt{d+e x} (-2 b e g+c d g+3 c e f)}{8 e^2 (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{35 c^2 (-2 b e g+c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{11/2}}-\frac{35 c (-2 b e g+c d g+3 c e f)}{24 e^2 \sqrt{d+e x} (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{7 c \sqrt{d+e x} (-2 b e g+c d g+3 c e f)}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{-2 b e g+c d g+3 c e f}{4 e^2 \sqrt{d+e x} (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{e f-d g}{3 e^2 (d+e x)^{3/2} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(f + g*x)/((d + e*x)^(3/2)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
-(e*f - d*g)/(3*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (3*c*e*f + c*
d*g - 2*b*e*g)/(4*e^2*(2*c*d - b*e)^2*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (7*c*(3*c*e
*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x])/(12*e^2*(2*c*d - b*e)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (3
5*c*(3*c*e*f + c*d*g - 2*b*e*g))/(24*e^2*(2*c*d - b*e)^4*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2]) + (35*c^2*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x])/(8*e^2*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2]) - (35*c^2*(3*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])])/(8*e^2*(2*c*d - b*e)^(11/2))
Rule 792
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]
Rule 672
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)), I
nt[(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] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]
Rule 666
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 660
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{f+g x}{(d+e x)^{3/2} \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}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(3 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 )^{5/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{3 c e f+c d g-2 b e g}{4 e^2 (2 c d-b e)^2 \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 (3 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 )^{5/2}} \, dx}{8 e (2 c d-b e)^2}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{3 c e f+c d g-2 b e g}{4 e^2 (2 c d-b e)^2 \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 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(35 c (3 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}{24 e (2 c d-b e)^3}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{3 c e f+c d g-2 b e g}{4 e^2 (2 c d-b e)^2 \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 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{35 c (3 c e f+c d g-2 b e g)}{24 e^2 (2 c d-b e)^4 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\left (35 c^2 (3 c e f+c d g-2 b e g)\right ) \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}{16 e (2 c d-b e)^4}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{3 c e f+c d g-2 b e g}{4 e^2 (2 c d-b e)^2 \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 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{35 c (3 c e f+c d g-2 b e g)}{24 e^2 (2 c d-b e)^4 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{35 c^2 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{8 e^2 (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\left (35 c^2 (3 c e f+c d g-2 b e g)\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 e (2 c d-b e)^5}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{3 c e f+c d g-2 b e g}{4 e^2 (2 c d-b e)^2 \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 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{35 c (3 c e f+c d g-2 b e g)}{24 e^2 (2 c d-b e)^4 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{35 c^2 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{8 e^2 (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\left (35 c^2 (3 c e f+c d g-2 b e g)\right ) \operatorname{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 )}{8 (2 c d-b e)^5}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{3 c e f+c d g-2 b e g}{4 e^2 (2 c d-b e)^2 \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 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{12 e^2 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{35 c (3 c e f+c d g-2 b e g)}{24 e^2 (2 c d-b e)^4 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{35 c^2 (3 c e f+c d g-2 b e g) \sqrt{d+e x}}{8 e^2 (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{35 c^2 (3 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 )}{8 e^2 (2 c d-b e)^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.136245, size = 128, normalized size = 0.28 \[ \frac{\frac{3 c^2 (d+e x)^3 (-2 b e g+c d g+3 c e f) \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )}{(2 c d-b e)^3}+3 d g-3 e f}{9 e^2 (d+e x)^{3/2} (2 c d-b e) ((d+e x) (c (d-e x)-b e))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)/((d + e*x)^(3/2)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
(-3*e*f + 3*d*g + (3*c^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^3*Hypergeometric2F1[-3/2, 3, -1/2, (-(c*d) + b*
e + c*e*x)/(-2*c*d + b*e)])/(2*c*d - b*e)^3)/(9*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*((d + e*x)*(-(b*e) + c*(d -
e*x)))^(3/2))
________________________________________________________________________________________
Maple [B] time = 0.039, size = 2011, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)/(e*x+d)^(3/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
[Out]
-1/24*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^5*g*(
-c*e*x-b*e+c*d)^(1/2)-420*(b*e-2*c*d)^(1/2)*x^3*c^4*d*e^4*f+42*(b*e-2*c*d)^(1/2)*x^2*b^3*c*e^5*g-63*(b*e-2*c*d
)^(1/2)*x^2*b^2*c^2*e^5*f+126*(b*e-2*c*d)^(1/2)*x^2*c^4*d^3*e^2*g+378*(b*e-2*c*d)^(1/2)*x^2*c^4*d^2*e^3*f+18*(
b*e-2*c*d)^(1/2)*x*b^3*c*e^5*f+204*(b*e-2*c*d)^(1/2)*x*c^4*d^4*e*g+612*(b*e-2*c*d)^(1/2)*x*c^4*d^3*e^2*f-105*(
b*e-2*c*d)^(1/2)*x^4*c^4*d*e^4*g+280*(b*e-2*c*d)^(1/2)*x^3*b^2*c^2*e^5*g-420*(b*e-2*c*d)^(1/2)*x^3*b*c^3*e^5*f
-140*(b*e-2*c*d)^(1/2)*x^3*c^4*d^2*e^3*g+315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*e*f*(-c*
e*x-b*e+c*d)^(1/2)-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*c^4*e^5*f*(-c*e*x-b*e+c*d)^(1/2)+2
10*(b*e-2*c*d)^(1/2)*x^4*b*c^3*e^5*g+103*(b*e-2*c*d)^(1/2)*b^2*c^2*d^3*e^2*g-363*(b*e-2*c*d)^(1/2)*b^2*c^2*d^2
*e^3*f-326*(b*e-2*c*d)^(1/2)*b*c^3*d^4*e*g+304*(b*e-2*c*d)^(1/2)*b*c^3*d^3*e^2*f+56*(b*e-2*c*d)^(1/2)*b^3*c*d^
2*e^3*g+82*(b*e-2*c*d)^(1/2)*b^3*c*d*e^4*f-8*(b*e-2*c*d)^(1/2)*b^4*e^5*f+171*(b*e-2*c*d)^(1/2)*c^4*d^5*g+630*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b^2*c^2*d*e^4*g*(-c*e*x-b*e+c*d)^(1/2)-315*arctan((-c*e*x-
b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^3*d^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)-945*arctan((-c*e*x-b*e+c*d)^(1/2)
/(b*e-2*c*d)^(1/2))*x^2*b*c^3*d*e^4*f*(-c*e*x-b*e+c*d)^(1/2)+630*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/
2))*x*b^2*c^2*d^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)-735*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^3*d^
3*e^2*g*(-c*e*x-b*e+c*d)^(1/2)-945*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^3*d^2*e^3*f*(-c*e*x-
b*e+c*d)^(1/2)-315*(b*e-2*c*d)^(1/2)*x^4*c^4*e^5*f-12*(b*e-2*c*d)^(1/2)*x*b^4*e^5*g+(b*e-2*c*d)^(1/2)*c^4*d^4*
e*f-4*(b*e-2*c*d)^(1/2)*b^4*d*e^4*g+651*(b*e-2*c*d)^(1/2)*x^2*b^2*c^2*d*e^4*g-588*(b*e-2*c*d)^(1/2)*x^2*b*c^3*
d^2*e^3*g-1008*(b*e-2*c*d)^(1/2)*x^2*b*c^3*d*e^4*f+162*(b*e-2*c*d)^(1/2)*x*b^3*c*d*e^4*g+630*arctan((-c*e*x-b*
e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^4*d^3*e^2*f*(-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^2*c^2*d^3*e^2*g*(-c*e*x-b*e+c*d)^(1/2)-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*
c^3*d^4*e*g*(-c*e*x-b*e+c*d)^(1/2)-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^3*e^2*f*(-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))*x^4*b*c^3*e^5*g*(-c*e*x-b*e+c*d)^(1/2)-1
05*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*c^4*d*e^4*g*(-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))*x^3*b^2*c^2*e^5*g*(-c*e*x-b*e+c*d)^(1/2)-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(
b*e-2*c*d)^(1/2))*x^3*b*c^3*e^5*f*(-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))*
x^3*c^4*d^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)-630*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^4*d*e^4*f*
(-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))*x*c^4*d^4*e*g*(-c*e*x-b*e+c*d)^(1/
2)+282*(b*e-2*c*d)^(1/2)*x*b^2*c^2*d^2*e^3*g-234*(b*e-2*c*d)^(1/2)*x*b^2*c^2*d*e^4*f-588*(b*e-2*c*d)^(1/2)*x*b
*c^3*d^3*e^2*g-540*(b*e-2*c*d)^(1/2)*x*b*c^3*d^2*e^3*f+140*(b*e-2*c*d)^(1/2)*x^3*b*c^3*d*e^4*g+315*arctan((-c*
e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*b*c^3*d*e^4*g*(-c*e*x-b*e+c*d)^(1/2))/(e*x+d)^(7/2)/(c*e*x+b*e-c*d)^
2/e^2/(b*e-2*c*d)^(11/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)^(3/2)/(-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*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^(3/2)), x)
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Fricas [B] time = 2.90888, size = 8442, normalized size = 18.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)^(3/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")
[Out]
[1/48*(105*((3*c^5*e^7*f + (c^5*d*e^6 - 2*b*c^4*e^7)*g)*x^6 + 2*(3*(c^5*d*e^6 + b*c^4*e^7)*f + (c^5*d^2*e^5 -
b*c^4*d*e^6 - 2*b^2*c^3*e^7)*g)*x^5 - (3*(c^5*d^2*e^5 - 6*b*c^4*d*e^6 - b^2*c^3*e^7)*f + (c^5*d^3*e^4 - 8*b*c^
4*d^2*e^5 + 11*b^2*c^3*d*e^6 + 2*b^3*c^2*e^7)*g)*x^4 - 4*(3*(c^5*d^3*e^4 - b*c^4*d^2*e^5 - b^2*c^3*d*e^6)*f +
(c^5*d^4*e^3 - 3*b*c^4*d^3*e^4 + b^2*c^3*d^2*e^5 + 2*b^3*c^2*d*e^6)*g)*x^3 - (3*(c^5*d^4*e^3 + 4*b*c^4*d^3*e^4
- 6*b^2*c^3*d^2*e^5)*f + (c^5*d^5*e^2 + 2*b*c^4*d^4*e^3 - 14*b^2*c^3*d^3*e^4 + 12*b^3*c^2*d^2*e^5)*g)*x^2 + 3
*(c^5*d^6*e - 2*b*c^4*d^5*e^2 + b^2*c^3*d^4*e^3)*f + (c^5*d^7 - 4*b*c^4*d^6*e + 5*b^2*c^3*d^5*e^2 - 2*b^3*c^2*
d^4*e^3)*g + 2*(3*(c^5*d^5*e^2 - 3*b*c^4*d^4*e^3 + 2*b^2*c^3*d^3*e^4)*f + (c^5*d^6*e - 5*b*c^4*d^5*e^2 + 8*b^2
*c^3*d^4*e^3 - 4*b^3*c^2*d^3*e^4)*g)*x)*sqrt(2*c*d - b*e)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e
^2)*x + 2*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))/(e^2*x^2 + 2*d*e*x + d^2
)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(105*(3*(2*c^5*d*e^5 - b*c^4*e^6)*f + (2*c^5*d^2*e^4 - 5*b*c
^4*d*e^5 + 2*b^2*c^3*e^6)*g)*x^4 + 140*(3*(2*c^5*d^2*e^4 + b*c^4*d*e^5 - b^2*c^3*e^6)*f + (2*c^5*d^3*e^3 - 3*b
*c^4*d^2*e^4 - 3*b^2*c^3*d*e^5 + 2*b^3*c^2*e^6)*g)*x^3 - 21*(3*(12*c^5*d^3*e^3 - 38*b*c^4*d^2*e^4 + 14*b^2*c^3
*d*e^5 + b^3*c^2*e^6)*f + (12*c^5*d^4*e^2 - 62*b*c^4*d^3*e^3 + 90*b^2*c^3*d^2*e^4 - 27*b^3*c^2*d*e^5 - 2*b^4*c
*e^6)*g)*x^2 - (2*c^5*d^5*e + 607*b*c^4*d^4*e^2 - 1030*b^2*c^3*d^3*e^3 + 527*b^3*c^2*d^2*e^4 - 98*b^4*c*d*e^5
+ 8*b^5*e^6)*f - (342*c^5*d^6 - 823*b*c^4*d^5*e + 532*b^2*c^3*d^4*e^2 + 9*b^3*c^2*d^3*e^3 - 64*b^4*c*d^2*e^4 +
4*b^5*d*e^5)*g - 6*(3*(68*c^5*d^4*e^2 - 94*b*c^4*d^3*e^3 + 4*b^2*c^3*d^2*e^4 + 15*b^3*c^2*d*e^5 - b^4*c*e^6)*
f + (68*c^5*d^5*e - 230*b*c^4*d^4*e^2 + 192*b^2*c^3*d^3*e^3 + 7*b^3*c^2*d^2*e^4 - 31*b^4*c*d*e^5 + 2*b^5*e^6)*
g)*x)*sqrt(e*x + d))/(64*c^8*d^12*e^2 - 320*b*c^7*d^11*e^3 + 688*b^2*c^6*d^10*e^4 - 832*b^3*c^5*d^9*e^5 + 620*
b^4*c^4*d^8*e^6 - 292*b^5*c^3*d^7*e^7 + 85*b^6*c^2*d^6*e^8 - 14*b^7*c*d^5*e^9 + b^8*d^4*e^10 + (64*c^8*d^6*e^8
- 192*b*c^7*d^5*e^9 + 240*b^2*c^6*d^4*e^10 - 160*b^3*c^5*d^3*e^11 + 60*b^4*c^4*d^2*e^12 - 12*b^5*c^3*d*e^13 +
b^6*c^2*e^14)*x^6 + 2*(64*c^8*d^7*e^7 - 128*b*c^7*d^6*e^8 + 48*b^2*c^6*d^5*e^9 + 80*b^3*c^5*d^4*e^10 - 100*b^
4*c^4*d^3*e^11 + 48*b^5*c^3*d^2*e^12 - 11*b^6*c^2*d*e^13 + b^7*c*e^14)*x^5 - (64*c^8*d^8*e^6 - 576*b*c^7*d^7*e
^7 + 1328*b^2*c^6*d^6*e^8 - 1408*b^3*c^5*d^5*e^9 + 780*b^4*c^4*d^4*e^10 - 212*b^5*c^3*d^3*e^11 + 13*b^6*c^2*d^
2*e^12 + 6*b^7*c*d*e^13 - b^8*e^14)*x^4 - 4*(64*c^8*d^9*e^5 - 256*b*c^7*d^8*e^6 + 368*b^2*c^6*d^7*e^7 - 208*b^
3*c^5*d^6*e^8 - 20*b^4*c^4*d^5*e^9 + 88*b^5*c^3*d^4*e^10 - 47*b^6*c^2*d^3*e^11 + 11*b^7*c*d^2*e^12 - b^8*d*e^1
3)*x^3 - (64*c^8*d^10*e^4 + 64*b*c^7*d^9*e^5 - 912*b^2*c^6*d^8*e^6 + 1952*b^3*c^5*d^7*e^7 - 2020*b^4*c^4*d^6*e
^8 + 1188*b^5*c^3*d^5*e^9 - 407*b^6*c^2*d^4*e^10 + 76*b^7*c*d^3*e^11 - 6*b^8*d^2*e^12)*x^2 + 2*(64*c^8*d^11*e^
3 - 384*b*c^7*d^10*e^4 + 944*b^2*c^6*d^9*e^5 - 1264*b^3*c^5*d^8*e^6 + 1020*b^4*c^4*d^7*e^7 - 512*b^5*c^3*d^6*e
^8 + 157*b^6*c^2*d^5*e^9 - 27*b^7*c*d^4*e^10 + 2*b^8*d^3*e^11)*x), -1/24*(105*((3*c^5*e^7*f + (c^5*d*e^6 - 2*b
*c^4*e^7)*g)*x^6 + 2*(3*(c^5*d*e^6 + b*c^4*e^7)*f + (c^5*d^2*e^5 - b*c^4*d*e^6 - 2*b^2*c^3*e^7)*g)*x^5 - (3*(c
^5*d^2*e^5 - 6*b*c^4*d*e^6 - b^2*c^3*e^7)*f + (c^5*d^3*e^4 - 8*b*c^4*d^2*e^5 + 11*b^2*c^3*d*e^6 + 2*b^3*c^2*e^
7)*g)*x^4 - 4*(3*(c^5*d^3*e^4 - b*c^4*d^2*e^5 - b^2*c^3*d*e^6)*f + (c^5*d^4*e^3 - 3*b*c^4*d^3*e^4 + b^2*c^3*d^
2*e^5 + 2*b^3*c^2*d*e^6)*g)*x^3 - (3*(c^5*d^4*e^3 + 4*b*c^4*d^3*e^4 - 6*b^2*c^3*d^2*e^5)*f + (c^5*d^5*e^2 + 2*
b*c^4*d^4*e^3 - 14*b^2*c^3*d^3*e^4 + 12*b^3*c^2*d^2*e^5)*g)*x^2 + 3*(c^5*d^6*e - 2*b*c^4*d^5*e^2 + b^2*c^3*d^4
*e^3)*f + (c^5*d^7 - 4*b*c^4*d^6*e + 5*b^2*c^3*d^5*e^2 - 2*b^3*c^2*d^4*e^3)*g + 2*(3*(c^5*d^5*e^2 - 3*b*c^4*d^
4*e^3 + 2*b^2*c^3*d^3*e^4)*f + (c^5*d^6*e - 5*b*c^4*d^5*e^2 + 8*b^2*c^3*d^4*e^3 - 4*b^3*c^2*d^3*e^4)*g)*x)*sqr
t(-2*c*d + b*e)*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)) + sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(105*(3*(2*c^5*d*e^5 - b*c^4*e^6)*f +
(2*c^5*d^2*e^4 - 5*b*c^4*d*e^5 + 2*b^2*c^3*e^6)*g)*x^4 + 140*(3*(2*c^5*d^2*e^4 + b*c^4*d*e^5 - b^2*c^3*e^6)*f
+ (2*c^5*d^3*e^3 - 3*b*c^4*d^2*e^4 - 3*b^2*c^3*d*e^5 + 2*b^3*c^2*e^6)*g)*x^3 - 21*(3*(12*c^5*d^3*e^3 - 38*b*c
^4*d^2*e^4 + 14*b^2*c^3*d*e^5 + b^3*c^2*e^6)*f + (12*c^5*d^4*e^2 - 62*b*c^4*d^3*e^3 + 90*b^2*c^3*d^2*e^4 - 27*
b^3*c^2*d*e^5 - 2*b^4*c*e^6)*g)*x^2 - (2*c^5*d^5*e + 607*b*c^4*d^4*e^2 - 1030*b^2*c^3*d^3*e^3 + 527*b^3*c^2*d^
2*e^4 - 98*b^4*c*d*e^5 + 8*b^5*e^6)*f - (342*c^5*d^6 - 823*b*c^4*d^5*e + 532*b^2*c^3*d^4*e^2 + 9*b^3*c^2*d^3*e
^3 - 64*b^4*c*d^2*e^4 + 4*b^5*d*e^5)*g - 6*(3*(68*c^5*d^4*e^2 - 94*b*c^4*d^3*e^3 + 4*b^2*c^3*d^2*e^4 + 15*b^3*
c^2*d*e^5 - b^4*c*e^6)*f + (68*c^5*d^5*e - 230*b*c^4*d^4*e^2 + 192*b^2*c^3*d^3*e^3 + 7*b^3*c^2*d^2*e^4 - 31*b^
4*c*d*e^5 + 2*b^5*e^6)*g)*x)*sqrt(e*x + d))/(64*c^8*d^12*e^2 - 320*b*c^7*d^11*e^3 + 688*b^2*c^6*d^10*e^4 - 832
*b^3*c^5*d^9*e^5 + 620*b^4*c^4*d^8*e^6 - 292*b^5*c^3*d^7*e^7 + 85*b^6*c^2*d^6*e^8 - 14*b^7*c*d^5*e^9 + b^8*d^4
*e^10 + (64*c^8*d^6*e^8 - 192*b*c^7*d^5*e^9 + 240*b^2*c^6*d^4*e^10 - 160*b^3*c^5*d^3*e^11 + 60*b^4*c^4*d^2*e^1
2 - 12*b^5*c^3*d*e^13 + b^6*c^2*e^14)*x^6 + 2*(64*c^8*d^7*e^7 - 128*b*c^7*d^6*e^8 + 48*b^2*c^6*d^5*e^9 + 80*b^
3*c^5*d^4*e^10 - 100*b^4*c^4*d^3*e^11 + 48*b^5*c^3*d^2*e^12 - 11*b^6*c^2*d*e^13 + b^7*c*e^14)*x^5 - (64*c^8*d^
8*e^6 - 576*b*c^7*d^7*e^7 + 1328*b^2*c^6*d^6*e^8 - 1408*b^3*c^5*d^5*e^9 + 780*b^4*c^4*d^4*e^10 - 212*b^5*c^3*d
^3*e^11 + 13*b^6*c^2*d^2*e^12 + 6*b^7*c*d*e^13 - b^8*e^14)*x^4 - 4*(64*c^8*d^9*e^5 - 256*b*c^7*d^8*e^6 + 368*b
^2*c^6*d^7*e^7 - 208*b^3*c^5*d^6*e^8 - 20*b^4*c^4*d^5*e^9 + 88*b^5*c^3*d^4*e^10 - 47*b^6*c^2*d^3*e^11 + 11*b^7
*c*d^2*e^12 - b^8*d*e^13)*x^3 - (64*c^8*d^10*e^4 + 64*b*c^7*d^9*e^5 - 912*b^2*c^6*d^8*e^6 + 1952*b^3*c^5*d^7*e
^7 - 2020*b^4*c^4*d^6*e^8 + 1188*b^5*c^3*d^5*e^9 - 407*b^6*c^2*d^4*e^10 + 76*b^7*c*d^3*e^11 - 6*b^8*d^2*e^12)*
x^2 + 2*(64*c^8*d^11*e^3 - 384*b*c^7*d^10*e^4 + 944*b^2*c^6*d^9*e^5 - 1264*b^3*c^5*d^8*e^6 + 1020*b^4*c^4*d^7*
e^7 - 512*b^5*c^3*d^6*e^8 + 157*b^6*c^2*d^5*e^9 - 27*b^7*c*d^4*e^10 + 2*b^8*d^3*e^11)*x)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)**(3/2)/(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)^(3/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
[Out]
[undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, 1]