3.20.98 \(\int \frac {f+g x}{(d+e x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=208 \[ -\frac {2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 208, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used =
{792, 614, 613} \begin {gather*} -\frac {2 (e f-d g)}{5 e^2 (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 {16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
(2*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
)) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (16*c*(8*c*
e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
Rule 613
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 614
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]
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]
Rubi steps
\begin {align*} \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=-\frac {2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(8 c e f+2 c d g-5 b e g) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{5 e (2 c d-b e)}\\ &=\frac {2 (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (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 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(8 c (8 c e f+2 c d g-5 b e g)) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{15 e (2 c d-b e)^3}\\ &=\frac {2 (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (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 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {16 c (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 345, normalized size = 1.66 \begin {gather*} -\frac {2 \left (b^4 e^4 (2 d g+3 e f+5 e g x)-2 b^3 c e^3 \left (19 d^2 g+2 d e (8 f+23 g x)+e^2 x (4 f+15 g x)\right )+12 b^2 c^2 e^2 \left (d^3 g+2 d^2 e (7 f-g x)+d e^2 x (12 f-19 g x)+2 e^3 x^2 (2 f-5 g x)\right )+8 b c^3 e \left (9 d^4 g-6 d^3 e (4 f-3 g x)+3 d^2 e^2 x (4 f+9 g x)-4 d e^3 x^2 (g x-12 f)+2 e^4 x^3 (12 f-5 g x)\right )-16 c^4 \left (3 d^5 g-3 d^4 e (f-g x)+3 d^3 e^2 x (4 f+g x)-2 d^2 e^3 x^2 (g x-6 f)-2 d e^4 x^3 (4 f+g x)-8 e^5 f x^4\right )\right )}{15 e^2 (d+e x)^2 (b e-2 c d)^5 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
(-2*(b^4*e^4*(3*e*f + 2*d*g + 5*e*g*x) + 12*b^2*c^2*e^2*(d^3*g + d*e^2*x*(12*f - 19*g*x) + 2*e^3*x^2*(2*f - 5*
g*x) + 2*d^2*e*(7*f - g*x)) - 16*c^4*(3*d^5*g - 8*e^5*f*x^4 - 3*d^4*e*(f - g*x) - 2*d^2*e^3*x^2*(-6*f + g*x) +
3*d^3*e^2*x*(4*f + g*x) - 2*d*e^4*x^3*(4*f + g*x)) + 8*b*c^3*e*(9*d^4*g + 2*e^4*x^3*(12*f - 5*g*x) - 6*d^3*e*
(4*f - 3*g*x) - 4*d*e^3*x^2*(-12*f + g*x) + 3*d^2*e^2*x*(4*f + 9*g*x)) - 2*b^3*c*e^3*(19*d^2*g + e^2*x*(4*f +
15*g*x) + 2*d*e*(8*f + 23*g*x))))/(15*e^2*(-2*c*d + b*e)^5*(d + e*x)^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(
-(b*e) + c*(d - e*x))])
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IntegrateAlgebraic [F] time = 180.26, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
$Aborted
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fricas [B] time = 93.10, size = 1028, normalized size = 4.94 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (16 \, {\left (8 \, c^{4} e^{5} f + {\left (2 \, c^{4} d e^{4} - 5 \, b c^{3} e^{5}\right )} g\right )} x^{4} + 8 \, {\left (8 \, {\left (2 \, c^{4} d e^{4} + 3 \, b c^{3} e^{5}\right )} f + {\left (4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} - 15 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} - 6 \, {\left (8 \, {\left (4 \, c^{4} d^{2} e^{3} - 8 \, b c^{3} d e^{4} - b^{2} c^{2} e^{5}\right )} f + {\left (8 \, c^{4} d^{3} e^{2} - 36 \, b c^{3} d^{2} e^{3} + 38 \, b^{2} c^{2} d e^{4} + 5 \, b^{3} c e^{5}\right )} g\right )} x^{2} + {\left (48 \, c^{4} d^{4} e - 192 \, b c^{3} d^{3} e^{2} + 168 \, b^{2} c^{2} d^{2} e^{3} - 32 \, b^{3} c d e^{4} + 3 \, b^{4} e^{5}\right )} f - 2 \, {\left (24 \, c^{4} d^{5} - 36 \, b c^{3} d^{4} e - 6 \, b^{2} c^{2} d^{3} e^{2} + 19 \, b^{3} c d^{2} e^{3} - b^{4} d e^{4}\right )} g - {\left (8 \, {\left (24 \, c^{4} d^{3} e^{2} - 12 \, b c^{3} d^{2} e^{3} - 18 \, b^{2} c^{2} d e^{4} + b^{3} c e^{5}\right )} f + {\left (48 \, c^{4} d^{4} e - 144 \, b c^{3} d^{3} e^{2} + 24 \, b^{2} c^{2} d^{2} e^{3} + 92 \, b^{3} c d e^{4} - 5 \, b^{4} e^{5}\right )} g\right )} x\right )}}{15 \, {\left (32 \, c^{7} d^{10} e^{2} - 144 \, b c^{6} d^{9} e^{3} + 272 \, b^{2} c^{5} d^{8} e^{4} - 280 \, b^{3} c^{4} d^{7} e^{5} + 170 \, b^{4} c^{3} d^{6} e^{6} - 61 \, b^{5} c^{2} d^{5} e^{7} + 12 \, b^{6} c d^{4} e^{8} - b^{7} d^{3} e^{9} + {\left (32 \, c^{7} d^{5} e^{7} - 80 \, b c^{6} d^{4} e^{8} + 80 \, b^{2} c^{5} d^{3} e^{9} - 40 \, b^{3} c^{4} d^{2} e^{10} + 10 \, b^{4} c^{3} d e^{11} - b^{5} c^{2} e^{12}\right )} x^{5} + {\left (32 \, c^{7} d^{6} e^{6} - 16 \, b c^{6} d^{5} e^{7} - 80 \, b^{2} c^{5} d^{4} e^{8} + 120 \, b^{3} c^{4} d^{3} e^{9} - 70 \, b^{4} c^{3} d^{2} e^{10} + 19 \, b^{5} c^{2} d e^{11} - 2 \, b^{6} c e^{12}\right )} x^{4} - {\left (64 \, c^{7} d^{7} e^{5} - 288 \, b c^{6} d^{6} e^{6} + 448 \, b^{2} c^{5} d^{5} e^{7} - 320 \, b^{3} c^{4} d^{4} e^{8} + 100 \, b^{4} c^{3} d^{3} e^{9} - 2 \, b^{5} c^{2} d^{2} e^{10} - 6 \, b^{6} c d e^{11} + b^{7} e^{12}\right )} x^{3} - {\left (64 \, c^{7} d^{8} e^{4} - 160 \, b c^{6} d^{7} e^{5} + 64 \, b^{2} c^{5} d^{6} e^{6} + 160 \, b^{3} c^{4} d^{5} e^{7} - 220 \, b^{4} c^{3} d^{4} e^{8} + 118 \, b^{5} c^{2} d^{3} e^{9} - 30 \, b^{6} c d^{2} e^{10} + 3 \, b^{7} d e^{11}\right )} x^{2} + {\left (32 \, c^{7} d^{9} e^{3} - 208 \, b c^{6} d^{8} e^{4} + 496 \, b^{2} c^{5} d^{7} e^{5} - 600 \, b^{3} c^{4} d^{6} e^{6} + 410 \, b^{4} c^{3} d^{5} e^{7} - 161 \, b^{5} c^{2} d^{4} e^{8} + 34 \, b^{6} c d^{3} e^{9} - 3 \, b^{7} d^{2} e^{10}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")
[Out]
-2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(16*(8*c^4*e^5*f + (2*c^4*d*e^4 - 5*b*c^3*e^5)*g)*x^4 + 8*(8*
(2*c^4*d*e^4 + 3*b*c^3*e^5)*f + (4*c^4*d^2*e^3 - 4*b*c^3*d*e^4 - 15*b^2*c^2*e^5)*g)*x^3 - 6*(8*(4*c^4*d^2*e^3
- 8*b*c^3*d*e^4 - b^2*c^2*e^5)*f + (8*c^4*d^3*e^2 - 36*b*c^3*d^2*e^3 + 38*b^2*c^2*d*e^4 + 5*b^3*c*e^5)*g)*x^2
+ (48*c^4*d^4*e - 192*b*c^3*d^3*e^2 + 168*b^2*c^2*d^2*e^3 - 32*b^3*c*d*e^4 + 3*b^4*e^5)*f - 2*(24*c^4*d^5 - 36
*b*c^3*d^4*e - 6*b^2*c^2*d^3*e^2 + 19*b^3*c*d^2*e^3 - b^4*d*e^4)*g - (8*(24*c^4*d^3*e^2 - 12*b*c^3*d^2*e^3 - 1
8*b^2*c^2*d*e^4 + b^3*c*e^5)*f + (48*c^4*d^4*e - 144*b*c^3*d^3*e^2 + 24*b^2*c^2*d^2*e^3 + 92*b^3*c*d*e^4 - 5*b
^4*e^5)*g)*x)/(32*c^7*d^10*e^2 - 144*b*c^6*d^9*e^3 + 272*b^2*c^5*d^8*e^4 - 280*b^3*c^4*d^7*e^5 + 170*b^4*c^3*d
^6*e^6 - 61*b^5*c^2*d^5*e^7 + 12*b^6*c*d^4*e^8 - b^7*d^3*e^9 + (32*c^7*d^5*e^7 - 80*b*c^6*d^4*e^8 + 80*b^2*c^5
*d^3*e^9 - 40*b^3*c^4*d^2*e^10 + 10*b^4*c^3*d*e^11 - b^5*c^2*e^12)*x^5 + (32*c^7*d^6*e^6 - 16*b*c^6*d^5*e^7 -
80*b^2*c^5*d^4*e^8 + 120*b^3*c^4*d^3*e^9 - 70*b^4*c^3*d^2*e^10 + 19*b^5*c^2*d*e^11 - 2*b^6*c*e^12)*x^4 - (64*c
^7*d^7*e^5 - 288*b*c^6*d^6*e^6 + 448*b^2*c^5*d^5*e^7 - 320*b^3*c^4*d^4*e^8 + 100*b^4*c^3*d^3*e^9 - 2*b^5*c^2*d
^2*e^10 - 6*b^6*c*d*e^11 + b^7*e^12)*x^3 - (64*c^7*d^8*e^4 - 160*b*c^6*d^7*e^5 + 64*b^2*c^5*d^6*e^6 + 160*b^3*
c^4*d^5*e^7 - 220*b^4*c^3*d^4*e^8 + 118*b^5*c^2*d^3*e^9 - 30*b^6*c*d^2*e^10 + 3*b^7*d*e^11)*x^2 + (32*c^7*d^9*
e^3 - 208*b*c^6*d^8*e^4 + 496*b^2*c^5*d^7*e^5 - 600*b^3*c^4*d^6*e^6 + 410*b^4*c^3*d^5*e^7 - 161*b^5*c^2*d^4*e^
8 + 34*b^6*c*d^3*e^9 - 3*b^7*d^2*e^10)*x)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.55Unable to transpose Error: Bad Argument Value
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maple [B] time = 0.06, size = 557, normalized size = 2.68 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-80 b \,c^{3} e^{5} g \,x^{4}+32 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}-120 b^{2} c^{2} e^{5} g \,x^{3}-32 b \,c^{3} d \,e^{4} g \,x^{3}+192 b \,c^{3} e^{5} f \,x^{3}+32 c^{4} d^{2} e^{3} g \,x^{3}+128 c^{4} d \,e^{4} f \,x^{3}-30 b^{3} c \,e^{5} g \,x^{2}-228 b^{2} c^{2} d \,e^{4} g \,x^{2}+48 b^{2} c^{2} e^{5} f \,x^{2}+216 b \,c^{3} d^{2} e^{3} g \,x^{2}+384 b \,c^{3} d \,e^{4} f \,x^{2}-48 c^{4} d^{3} e^{2} g \,x^{2}-192 c^{4} d^{2} e^{3} f \,x^{2}+5 b^{4} e^{5} g x -92 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x -24 b^{2} c^{2} d^{2} e^{3} g x +144 b^{2} c^{2} d \,e^{4} f x +144 b \,c^{3} d^{3} e^{2} g x +96 b \,c^{3} d^{2} e^{3} f x -48 c^{4} d^{4} e g x -192 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +3 b^{4} e^{5} f -38 b^{3} c \,d^{2} e^{3} g -32 b^{3} c d \,e^{4} f +12 b^{2} c^{2} d^{3} e^{2} g +168 b^{2} c^{2} d^{2} e^{3} f +72 b \,c^{3} d^{4} e g -192 b \,c^{3} d^{3} e^{2} f -48 c^{4} d^{5} g +48 c^{4} d^{4} e f \right )}{15 \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} c^{2} d^{2} e^{3}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 c^{5} d^{5}\right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
[Out]
-2/15*(c*e*x+b*e-c*d)*(-80*b*c^3*e^5*g*x^4+32*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4-120*b^2*c^2*e^5*g*x^3-32*b*c^3
*d*e^4*g*x^3+192*b*c^3*e^5*f*x^3+32*c^4*d^2*e^3*g*x^3+128*c^4*d*e^4*f*x^3-30*b^3*c*e^5*g*x^2-228*b^2*c^2*d*e^4
*g*x^2+48*b^2*c^2*e^5*f*x^2+216*b*c^3*d^2*e^3*g*x^2+384*b*c^3*d*e^4*f*x^2-48*c^4*d^3*e^2*g*x^2-192*c^4*d^2*e^3
*f*x^2+5*b^4*e^5*g*x-92*b^3*c*d*e^4*g*x-8*b^3*c*e^5*f*x-24*b^2*c^2*d^2*e^3*g*x+144*b^2*c^2*d*e^4*f*x+144*b*c^3
*d^3*e^2*g*x+96*b*c^3*d^2*e^3*f*x-48*c^4*d^4*e*g*x-192*c^4*d^3*e^2*f*x+2*b^4*d*e^4*g+3*b^4*e^5*f-38*b^3*c*d^2*
e^3*g-32*b^3*c*d*e^4*f+12*b^2*c^2*d^3*e^2*g+168*b^2*c^2*d^2*e^3*f+72*b*c^3*d^4*e*g-192*b*c^3*d^3*e^2*f-48*c^4*
d^5*g+48*c^4*d^4*e*f)/(b^5*e^5-10*b^4*c*d*e^4+40*b^3*c^2*d^2*e^3-80*b^2*c^3*d^3*e^2+80*b*c^4*d^4*e-32*c^5*d^5)
/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?
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mupad [B] time = 5.13, size = 3326, normalized size = 15.99
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)/((d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2)),x)
[Out]
(x*((16*c^2*(b*g - c*f))/(15*(b*e - 2*c*d)^5) - (8*b*c^2*g)/(15*(b*e - 2*c*d)^5)) + (72*c^3*d*e*f - 56*c^3*d^2
*g - 44*b*c^2*e^2*f + 10*b^2*c*e^2*g + 20*b*c^2*d*e*g)/(15*e^2*(b*e - 2*c*d)^5) + (8*c^2*g*(c*d^2 - b*d*e))/(1
5*e^2*(b*e - 2*c*d)^5))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + (((4*b*c*g)/(15*e*(b*e - 2*c*d)^5) - (8*
c^2*d*g)/(15*e^2*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((2*e^2*f)/(5*b^3
*e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*d^2*e^4 - 30*b^2*c*d*e^5) - (2*d*e*g)/(5*b^3*e^6 - 40*c^3*d^3*e^3 + 60*b*c^2*
d^2*e^4 - 30*b^2*c*d*e^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((2*b*g)/(5*(3*b*e^2 -
6*c*d*e)*(b*e - 2*c*d)^3) - (4*c*d*g)/(5*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e -
b*e^2*x)^(1/2))/(d + e*x)^2 - (((4*c*g*(3*b*e - 4*c*d))/(15*e^2*(b*e - 2*c*d)^5) - (8*c^2*d*g)/(15*e^2*(b*e -
2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + ((x*(((e*(b*e - c*d) + c*d*e)*(((e*(b*e -
c*d) + c*d*e)*((4*c^4*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^4
*g*(e*(b*e - c*d) + c*d*e))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (4*b*c^4*e^2*g)/(15*(
b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(8*b^2*c*e^3*g - 26*b*c^2*e^3*f + 36*
c^3*d*e^2*f - 32*c^3*d^2*e*g + 14*b*c^2*d*e^2*g))/(15*e*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))
- (2*b*c^3*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (8*c^4*d*g*(b*e
- c*d))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (d*(b*e - c*d)*((4*c^4*e^2*(5*b
*g - 4*c*f))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^4*g*(e*(b*e - c*d) + c*d*e))/(1
5*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (4*b*c^4*e^2*g)/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^
2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(12*c^3*d^3*g - 12*b^3*e^3*g + 28*b^2*c*e^3*f + 68*c^3*d^2*e*f - 86
*b*c^2*d*e^2*f - 36*b*c^2*d^2*e*g + 36*b^2*c*d*e^2*g))/(15*e*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*
d*e)) + (b*c*(8*b^2*c*e^3*g - 26*b*c^2*e^3*f + 36*c^3*d*e^2*f - 32*c^3*d^2*e*g + 14*b*c^2*d*e^2*g))/(15*e*(b*e
- 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))) + (d*(b*e - c*d)*(((e*(b*e - c*d) + c*d*e)*((4*c^4*e^2*(5*
b*g - 4*c*f))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^4*g*(e*(b*e - c*d) + c*d*e))/(
15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (4*b*c^4*e^2*g)/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b
^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (2*c^2*(8*b^2*c*e^3*g - 26*b*c^2*e^3*f + 36*c^3*d*e^2*f - 32*c^3*d^2*e*g
+ 14*b*c^2*d*e^2*g))/(15*e*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (2*b*c^3*e^2*(5*b*g - 4*c*
f))/(15*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (8*c^4*d*g*(b*e - c*d))/(15*(b*e - 2*c*d)^3*(
4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (b*c*(12*c^3*d^3*g - 12*b^3*e^3*g + 28*b^2*c*e^3*f + 68*c^3*
d^2*e*f - 86*b*c^2*d*e^2*f - 36*b*c^2*d^2*e*g + 36*b^2*c*d*e^2*g))/(15*e*(b*e - 2*c*d)^3*(4*c^3*d^2 + b^2*c*e^
2 - 4*b*c^2*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/((d + e*x)^2*(b*e - c*d + c*e*x)^2) + ((x*(((e
*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((16*c^5*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 +
b^2*c*e^2 - 4*b*c^2*d*e)) - (32*c^5*g*(e*(b*e - c*d) + c*d*e))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4
*b*c^2*d*e)) + (16*b*c^5*e^2*g)/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*
(104*b*c^3*e^3*f - 48*b^2*c^2*e^3*g - 144*c^4*d*e^2*f + 64*c^4*d^2*e*g + 8*b*c^3*d*e^2*g))/(15*e*(b*e - 2*c*d)
^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^4*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2
*c*e^2 - 4*b*c^2*d*e)) + (32*c^5*d*g*(b*e - c*d))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))
/(c*e^2) - (d*(b*e - c*d)*((16*c^5*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d
*e)) - (32*c^5*g*(e*(b*e - c*d) + c*d*e))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*b*c
^5*e^2*g)/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*(96*c^4*d^3*g + 84*b^2
*c^2*e^3*f - 60*b^3*c*e^3*g + 512*c^4*d^2*e*f - 440*b*c^3*d*e^2*f - 432*b*c^3*d^2*e*g + 324*b^2*c^2*d*e^2*g))/
(15*e*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (b*c*(104*b*c^3*e^3*f - 48*b^2*c^2*e^3*g - 144*
c^4*d*e^2*f + 64*c^4*d^2*e*g + 8*b*c^3*d*e^2*g))/(15*e*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)))
+ (d*(b*e - c*d)*(((e*(b*e - c*d) + c*d*e)*((16*c^5*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2
*c*e^2 - 4*b*c^2*d*e)) - (32*c^5*g*(e*(b*e - c*d) + c*d*e))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c
^2*d*e)) + (16*b*c^5*e^2*g)/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*(104
*b*c^3*e^3*f - 48*b^2*c^2*e^3*g - 144*c^4*d*e^2*f + 64*c^4*d^2*e*g + 8*b*c^3*d*e^2*g))/(15*e*(b*e - 2*c*d)^5*(
4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^4*e^2*(5*b*g - 4*c*f))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e
^2 - 4*b*c^2*d*e)) + (32*c^5*d*g*(b*e - c*d))/(15*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*
e^2) + (b*c*(96*c^4*d^3*g + 84*b^2*c^2*e^3*f - 60*b^3*c*e^3*g + 512*c^4*d^2*e*f - 440*b*c^3*d*e^2*f - 432*b*c^
3*d^2*e*g + 324*b^2*c^2*d*e^2*g))/(15*e*(b*e - 2*c*d)^5*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)))*(c*d^2 - c*e^2
*x^2 - b*d*e - b*e^2*x)^(1/2))/((d + e*x)*(b*e - c*d + c*e*x))
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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}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)/(-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)*(d + e*x)), x)
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