3.20.92 \(\int \frac {f+g x}{(d+e x)^3 (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=284 \[ \frac {16 c^2 (b+2 c x) (-7 b e g+6 c d g+8 c e f)}{35 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {4 c (-7 b e g+6 c d g+8 c e f)}{35 e^2 (d+e x) (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (-7 b e g+6 c d g+8 c e f)}{35 e^2 (d+e x)^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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Rubi [A]  time = 0.39, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {792, 658, 613} \begin {gather*} \frac {16 c^2 (b+2 c x) (-7 b e g+6 c d g+8 c e f)}{35 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {4 c (-7 b e g+6 c d g+8 c e f)}{35 e^2 (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 {2 (-7 b e g+6 c d g+8 c e f)}{35 e^2 (d+e x)^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)^3*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

(16*c^2*(8*c*e*f + 6*c*d*g - 7*b*e*g)*(b + 2*c*x))/(35*e*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2]) - (2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(8*c*
e*f + 6*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (4*
c*(8*c*e*f + 6*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^3*(d + e*x)*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 658

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*Simplify[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, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

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)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(8 c e f+6 c d g-7 b e g) \int \frac {1}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{7 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (8 c e f+6 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(6 c (8 c e f+6 c d g-7 b e g)) \int \frac {1}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{35 e (2 c d-b e)^2}\\ &=-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (8 c e f+6 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {4 c (8 c e f+6 c d g-7 b e g)}{35 e^2 (2 c d-b e)^3 (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {\left (8 c^2 (8 c e f+6 c d g-7 b e g)\right ) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{35 e (2 c d-b e)^3}\\ &=\frac {16 c^2 (8 c e f+6 c d g-7 b e g) (b+2 c x)}{35 e (2 c d-b e)^5 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (8 c e f+6 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {4 c (8 c e f+6 c d g-7 b e g)}{35 e^2 (2 c d-b e)^3 (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 331, normalized size = 1.17 \begin {gather*} \frac {2 \left (b^4 e^4 (2 d g+5 e f+7 e g x)-2 b^3 c e^3 \left (11 d^2 g+d e (24 f+38 g x)+e^2 x (4 f+7 g x)\right )+4 b^2 c^2 e^2 \left (31 d^3 g+2 d^2 e (23 f+53 g x)+d e^2 x (20 f+59 g x)+2 e^3 x^2 (2 f+7 g x)\right )-8 b c^3 e \left (15 d^4 g+d^3 e (48 f+46 g x)+d^2 e^2 x (52 f-11 g x)+4 d e^3 x^2 (8 f-9 g x)+2 e^4 x^3 (4 f-7 g x)\right )+16 c^4 \left (d^5 g+d^4 e (13 f+3 g x)+d^3 e^2 x (4 f-15 g x)-2 d^2 e^3 x^2 (10 f+9 g x)-6 d e^4 x^3 (4 f+g x)-8 e^5 f x^4\right )\right )}{35 e^2 (d+e x)^3 (b e-2 c d)^5 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/((d + e*x)^3*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

(2*(b^4*e^4*(5*e*f + 2*d*g + 7*e*g*x) + 16*c^4*(d^5*g - 8*e^5*f*x^4 + d^3*e^2*x*(4*f - 15*g*x) - 6*d*e^4*x^3*(
4*f + g*x) + d^4*e*(13*f + 3*g*x) - 2*d^2*e^3*x^2*(10*f + 9*g*x)) - 2*b^3*c*e^3*(11*d^2*g + e^2*x*(4*f + 7*g*x
) + d*e*(24*f + 38*g*x)) - 8*b*c^3*e*(15*d^4*g + d^2*e^2*x*(52*f - 11*g*x) + 4*d*e^3*x^2*(8*f - 9*g*x) + 2*e^4
*x^3*(4*f - 7*g*x) + d^3*e*(48*f + 46*g*x)) + 4*b^2*c^2*e^2*(31*d^3*g + 2*e^3*x^2*(2*f + 7*g*x) + 2*d^2*e*(23*
f + 53*g*x) + d*e^2*x*(20*f + 59*g*x))))/(35*e^2*(-2*c*d + b*e)^5*(d + e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d -
e*x))])

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IntegrateAlgebraic [F]  time = 180.24, 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)^3*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

$Aborted

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fricas [B]  time = 137.52, size = 974, normalized size = 3.43 \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 (6 \, c^{4} d e^{4} - 7 \, b c^{3} e^{5}\right )} g\right )} x^{4} + 8 \, {\left (8 \, {\left (6 \, c^{4} d e^{4} + b c^{3} e^{5}\right )} f + {\left (36 \, c^{4} d^{2} e^{3} - 36 \, b c^{3} d e^{4} - 7 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} + 2 \, {\left (8 \, {\left (20 \, c^{4} d^{2} e^{3} + 16 \, b c^{3} d e^{4} - b^{2} c^{2} e^{5}\right )} f + {\left (120 \, c^{4} d^{3} e^{2} - 44 \, b c^{3} d^{2} e^{3} - 118 \, b^{2} c^{2} d e^{4} + 7 \, b^{3} c e^{5}\right )} g\right )} x^{2} - {\left (208 \, c^{4} d^{4} e - 384 \, b c^{3} d^{3} e^{2} + 184 \, b^{2} c^{2} d^{2} e^{3} - 48 \, b^{3} c d e^{4} + 5 \, b^{4} e^{5}\right )} f - 2 \, {\left (8 \, c^{4} d^{5} - 60 \, b c^{3} d^{4} e + 62 \, b^{2} c^{2} d^{3} e^{2} - 11 \, b^{3} c d^{2} e^{3} + b^{4} d e^{4}\right )} g - {\left (8 \, {\left (8 \, c^{4} d^{3} e^{2} - 52 \, b c^{3} d^{2} e^{3} + 10 \, b^{2} c^{2} d e^{4} - b^{3} c e^{5}\right )} f + {\left (48 \, c^{4} d^{4} e - 368 \, b c^{3} d^{3} e^{2} + 424 \, b^{2} c^{2} d^{2} e^{3} - 76 \, b^{3} c d e^{4} + 7 \, b^{4} e^{5}\right )} g\right )} x\right )}}{35 \, {\left (32 \, c^{6} d^{10} e^{2} - 112 \, b c^{5} d^{9} e^{3} + 160 \, b^{2} c^{4} d^{8} e^{4} - 120 \, b^{3} c^{3} d^{7} e^{5} + 50 \, b^{4} c^{2} d^{6} e^{6} - 11 \, b^{5} c d^{5} e^{7} + b^{6} d^{4} e^{8} - {\left (32 \, c^{6} d^{5} e^{7} - 80 \, b c^{5} d^{4} e^{8} + 80 \, b^{2} c^{4} d^{3} e^{9} - 40 \, b^{3} c^{3} d^{2} e^{10} + 10 \, b^{4} c^{2} d e^{11} - b^{5} c e^{12}\right )} x^{5} - {\left (96 \, c^{6} d^{6} e^{6} - 208 \, b c^{5} d^{5} e^{7} + 160 \, b^{2} c^{4} d^{4} e^{8} - 40 \, b^{3} c^{3} d^{3} e^{9} - 10 \, b^{4} c^{2} d^{2} e^{10} + 7 \, b^{5} c d e^{11} - b^{6} e^{12}\right )} x^{4} - 2 \, {\left (32 \, c^{6} d^{7} e^{5} - 16 \, b c^{5} d^{6} e^{6} - 80 \, b^{2} c^{4} d^{5} e^{7} + 120 \, b^{3} c^{3} d^{4} e^{8} - 70 \, b^{4} c^{2} d^{3} e^{9} + 19 \, b^{5} c d^{2} e^{10} - 2 \, b^{6} d e^{11}\right )} x^{3} + 2 \, {\left (32 \, c^{6} d^{8} e^{4} - 176 \, b c^{5} d^{7} e^{5} + 320 \, b^{2} c^{4} d^{6} e^{6} - 280 \, b^{3} c^{3} d^{5} e^{7} + 130 \, b^{4} c^{2} d^{4} e^{8} - 31 \, b^{5} c d^{3} e^{9} + 3 \, b^{6} d^{2} e^{10}\right )} x^{2} + {\left (96 \, c^{6} d^{9} e^{3} - 368 \, b c^{5} d^{8} e^{4} + 560 \, b^{2} c^{4} d^{7} e^{5} - 440 \, b^{3} c^{3} d^{6} e^{6} + 190 \, b^{4} c^{2} d^{5} e^{7} - 43 \, b^{5} c d^{4} e^{8} + 4 \, b^{6} d^{3} e^{9}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/35*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(16*(8*c^4*e^5*f + (6*c^4*d*e^4 - 7*b*c^3*e^5)*g)*x^4 + 8*(8*(
6*c^4*d*e^4 + b*c^3*e^5)*f + (36*c^4*d^2*e^3 - 36*b*c^3*d*e^4 - 7*b^2*c^2*e^5)*g)*x^3 + 2*(8*(20*c^4*d^2*e^3 +
16*b*c^3*d*e^4 - b^2*c^2*e^5)*f + (120*c^4*d^3*e^2 - 44*b*c^3*d^2*e^3 - 118*b^2*c^2*d*e^4 + 7*b^3*c*e^5)*g)*x
^2 - (208*c^4*d^4*e - 384*b*c^3*d^3*e^2 + 184*b^2*c^2*d^2*e^3 - 48*b^3*c*d*e^4 + 5*b^4*e^5)*f - 2*(8*c^4*d^5 -
60*b*c^3*d^4*e + 62*b^2*c^2*d^3*e^2 - 11*b^3*c*d^2*e^3 + b^4*d*e^4)*g - (8*(8*c^4*d^3*e^2 - 52*b*c^3*d^2*e^3
+ 10*b^2*c^2*d*e^4 - b^3*c*e^5)*f + (48*c^4*d^4*e - 368*b*c^3*d^3*e^2 + 424*b^2*c^2*d^2*e^3 - 76*b^3*c*d*e^4 +
7*b^4*e^5)*g)*x)/(32*c^6*d^10*e^2 - 112*b*c^5*d^9*e^3 + 160*b^2*c^4*d^8*e^4 - 120*b^3*c^3*d^7*e^5 + 50*b^4*c^
2*d^6*e^6 - 11*b^5*c*d^5*e^7 + b^6*d^4*e^8 - (32*c^6*d^5*e^7 - 80*b*c^5*d^4*e^8 + 80*b^2*c^4*d^3*e^9 - 40*b^3*
c^3*d^2*e^10 + 10*b^4*c^2*d*e^11 - b^5*c*e^12)*x^5 - (96*c^6*d^6*e^6 - 208*b*c^5*d^5*e^7 + 160*b^2*c^4*d^4*e^8
- 40*b^3*c^3*d^3*e^9 - 10*b^4*c^2*d^2*e^10 + 7*b^5*c*d*e^11 - b^6*e^12)*x^4 - 2*(32*c^6*d^7*e^5 - 16*b*c^5*d^
6*e^6 - 80*b^2*c^4*d^5*e^7 + 120*b^3*c^3*d^4*e^8 - 70*b^4*c^2*d^3*e^9 + 19*b^5*c*d^2*e^10 - 2*b^6*d*e^11)*x^3
+ 2*(32*c^6*d^8*e^4 - 176*b*c^5*d^7*e^5 + 320*b^2*c^4*d^6*e^6 - 280*b^3*c^3*d^5*e^7 + 130*b^4*c^2*d^4*e^8 - 31
*b^5*c*d^3*e^9 + 3*b^6*d^2*e^10)*x^2 + (96*c^6*d^9*e^3 - 368*b*c^5*d^8*e^4 + 560*b^2*c^4*d^7*e^5 - 440*b^3*c^3
*d^6*e^6 + 190*b^4*c^2*d^5*e^7 - 43*b^5*c*d^4*e^8 + 4*b^6*d^3*e^9)*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)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to transpose Error: Bad Argument Value

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maple [B]  time = 0.06, size = 564, normalized size = 1.99 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (112 b \,c^{3} e^{5} g \,x^{4}-96 c^{4} d \,e^{4} g \,x^{4}-128 c^{4} e^{5} f \,x^{4}+56 b^{2} c^{2} e^{5} g \,x^{3}+288 b \,c^{3} d \,e^{4} g \,x^{3}-64 b \,c^{3} e^{5} f \,x^{3}-288 c^{4} d^{2} e^{3} g \,x^{3}-384 c^{4} d \,e^{4} f \,x^{3}-14 b^{3} c \,e^{5} g \,x^{2}+236 b^{2} c^{2} d \,e^{4} g \,x^{2}+16 b^{2} c^{2} e^{5} f \,x^{2}+88 b \,c^{3} d^{2} e^{3} g \,x^{2}-256 b \,c^{3} d \,e^{4} f \,x^{2}-240 c^{4} d^{3} e^{2} g \,x^{2}-320 c^{4} d^{2} e^{3} f \,x^{2}+7 b^{4} e^{5} g x -76 b^{3} c d \,e^{4} g x -8 b^{3} c \,e^{5} f x +424 b^{2} c^{2} d^{2} e^{3} g x +80 b^{2} c^{2} d \,e^{4} f x -368 b \,c^{3} d^{3} e^{2} g x -416 b \,c^{3} d^{2} e^{3} f x +48 c^{4} d^{4} e g x +64 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +5 b^{4} e^{5} f -22 b^{3} c \,d^{2} e^{3} g -48 b^{3} c d \,e^{4} f +124 b^{2} c^{2} d^{3} e^{2} g +184 b^{2} c^{2} d^{2} e^{3} f -120 b \,c^{3} d^{4} e g -384 b \,c^{3} d^{3} e^{2} f +16 c^{4} d^{5} g +208 c^{4} d^{4} e f \right )}{35 \left (e x +d \right )^{2} \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 {3}{2}} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/35*(c*e*x+b*e-c*d)*(112*b*c^3*e^5*g*x^4-96*c^4*d*e^4*g*x^4-128*c^4*e^5*f*x^4+56*b^2*c^2*e^5*g*x^3+288*b*c^3
*d*e^4*g*x^3-64*b*c^3*e^5*f*x^3-288*c^4*d^2*e^3*g*x^3-384*c^4*d*e^4*f*x^3-14*b^3*c*e^5*g*x^2+236*b^2*c^2*d*e^4
*g*x^2+16*b^2*c^2*e^5*f*x^2+88*b*c^3*d^2*e^3*g*x^2-256*b*c^3*d*e^4*f*x^2-240*c^4*d^3*e^2*g*x^2-320*c^4*d^2*e^3
*f*x^2+7*b^4*e^5*g*x-76*b^3*c*d*e^4*g*x-8*b^3*c*e^5*f*x+424*b^2*c^2*d^2*e^3*g*x+80*b^2*c^2*d*e^4*f*x-368*b*c^3
*d^3*e^2*g*x-416*b*c^3*d^2*e^3*f*x+48*c^4*d^4*e*g*x+64*c^4*d^3*e^2*f*x+2*b^4*d*e^4*g+5*b^4*e^5*f-22*b^3*c*d^2*
e^3*g-48*b^3*c*d*e^4*f+124*b^2*c^2*d^3*e^2*g+184*b^2*c^2*d^2*e^3*f-120*b*c^3*d^4*e*g-384*b*c^3*d^3*e^2*f+16*c^
4*d^5*g+208*c^4*d^4*e*f)/(e*x+d)^2/(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)^(3/2)

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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)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),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 = 7.86, size = 4339, normalized size = 15.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)/((d + e*x)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2)),x)

[Out]

(((8*c*g*(2*b*e - 3*c*d))/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (8*c^2*d*g)/(35*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 - (((d*((d*((8*c^4*(2*c*d*g - 7*b*e*
g + 6*c*e*f))/(105*(b*e - 2*c*d)^7) + (16*c^5*d*g)/(105*(b*e - 2*c*d)^7)))/e + (64*c^5*d^2*g - 20*b^2*c^3*e^2*
g - 320*c^5*d*e*f + 112*b*c^4*e^2*f + 80*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^7)))/e - (2*b*c^2*(16*c^2*d^2*g - 1
1*b^2*e^2*g + 34*b*c*e^2*f - 80*c^2*d*e*f + 22*b*c*d*e*g))/(105*e*(b*e - 2*c*d)^7))*(c*d^2 - c*e^2*x^2 - b*d*e
- b*e^2*x)^(1/2))/(d + e*x) - (((4*b*c*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (8*c^2*d*g)/(35*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 - (((8*c^2*g*(3*b*e -
4*c*d))/(105*e^2*(b*e - 2*c*d)^5) - (16*c^3*d*g)/(105*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) - (((d*((48*c^4*f - 40*b*c^3*g)/(105*(b*e - 2*c*d)^6) + (16*c^4*d*g)/(105*e*(b*e - 2*c*d
)^6)))/e + (8*b*c^2*(2*b*g - 3*c*f))/(105*(b*e - 2*c*d)^6))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d +
e*x) + (((d*((48*c^4*d*g + 48*c^4*e*f - 64*b*c^3*e*g)/(105*e*(b*e - 2*c*d)^6) + (16*c^4*d*g)/(105*e*(b*e - 2*c
*d)^6)))/e + (96*c^4*d*f - 72*b*c^3*d*g - 72*b*c^3*e*f + 52*b^2*c^2*e*g)/(105*e*(b*e - 2*c*d)^6))*(c*d^2 - c*e
^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((2*b*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (4*c*d*g)/(7
*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((d*((8*
c^3*d*g - 24*c^3*e*f + 16*b*c^2*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4) - (8*c^3*d*g)/(35*(3*b*e^2 - 6*c
*d*e)*(b*e - 2*c*d)^4)))/e - (2*b*c*(3*b*e*g + 2*c*d*g - 6*c*e*f))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^4))*(
c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((2*d*g)/(7*b^2*e^4 + 28*c^2*d^2*e^2 - 28*b*c*d*e^3
) - (2*e*f)/(7*b^2*e^4 + 28*c^2*d^2*e^2 - 28*b*c*d*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x
)^4 + (((d*((2*c*e*(3*b*e*g + 4*c*d*g - 6*c*e*f))/(7*(b*e - 2*c*d)^2*(5*b^2*e^4 + 20*c^2*d^2*e^2 - 20*b*c*d*e^
3)) - (4*c^2*d*e*g)/(7*(b*e - 2*c*d)^2*(5*b^2*e^4 + 20*c^2*d^2*e^2 - 20*b*c*d*e^3))))/e - (16*b^2*e^2*g + 16*c
^2*d^2*g - 26*b*c*e^2*f + 40*c^2*d*e*f - 32*b*c*d*e*g)/(7*(b*e - 2*c*d)^2*(5*b^2*e^4 + 20*c^2*d^2*e^2 - 20*b*c
*d*e^3)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((248*c^4*d^3*g + 162*b^2*c^2*e^3*f - 10
8*b^3*c*e^3*g + 488*c^4*d^2*e*f - 580*b*c^3*d*e^2*f - 604*b*c^3*d^2*e*g + 464*b^2*c^2*d*e^2*g)/(35*(b*e - 2*c*
d)^4*(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3)) + (d*((d*((24*c^3*e^3*(b*g - c*f))/(35*(b*e - 2*c*d)^4*(3*b^
2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3)) - (8*c^4*d*e^2*g)/(35*(b*e - 2*c*d)^4*(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12
*b*c*d*e^3))))/e - (68*b*c^3*e^3*f - 22*b^2*c^2*e^3*g - 184*c^4*d*e^2*f + 24*c^4*d^2*e*g + 68*b*c^3*d*e^2*g)/(
35*(b*e - 2*c*d)^4*(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3))))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/
2))/(d + e*x)^2 - ((x*(((e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*(((e*(b*e - c*d) + c*d*e)*((16*c^7*e
^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (32*c^7*e*g*(e
*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (16*b*c^7*e^3*g)/(105*(b*
e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (8*c^6*e*(15*b^2*e^2*g + 44*c^2*d^2*g + 16*b*c
*e^2*f - 68*c^2*d*e*f - 22*b*c*d*e*g))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^6*
e^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (32*c^7*d*e*g
*(b*e - c*d))/(105*(b*e - 2*c*d)^6*(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^7*
e^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (32*c^7*e*g*(
e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (16*b*c^7*e^3*g)/(105*(b
*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*(216*b^2*c^4*e^4*f - 246*b^3*c^3*e^4*g
+ 1264*c^6*d^2*e^2*f + 592*c^6*d^3*e*g - 992*b*c^5*d*e^3*f - 1696*b*c^5*d^2*e^2*g + 1140*b^2*c^4*d*e^3*g))/(1
05*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (4*b*c^5*e*(15*b^2*e^2*g + 44*c^2*d^2*g + 16*b*c
*e^2*f - 68*c^2*d*e*f - 22*b*c*d*e*g))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) -
(d*(b*e - c*d)*(((e*(b*e - c*d) + c*d*e)*((16*c^7*e^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4*
c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (32*c^7*e*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b
^2*c*e^2 - 4*b*c^2*d*e)) - (16*b*c^7*e^3*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e
^2) + (8*c^6*e*(15*b^2*e^2*g + 44*c^2*d^2*g + 16*b*c*e^2*f - 68*c^2*d*e*f - 22*b*c*d*e*g))/(105*(b*e - 2*c*d)^
6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^6*e^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4
*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (32*c^7*d*e*g*(b*e - c*d))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2
- 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*(2112*c^6*d^4*g - 538*b^3*c^3*e^4*f + 452*b^4*c^2*e^4*g + 3264*c^6*d^3*e*f
- 6152*b*c^5*d^3*e*g - 5528*b*c^5*d^2*e^2*f + 3012*b^2*c^4*d*e^3*f - 2832*b^3*c^3*d*e^3*g + 6420*b^2*c^4*d^2*e
^2*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (b*c*(216*b^2*c^4*e^4*f - 246*b^3*c^3*e
^4*g + 1264*c^6*d^2*e^2*f + 592*c^6*d^3*e*g - 992*b*c^5*d*e^3*f - 1696*b*c^5*d^2*e^2*g + 1140*b^2*c^4*d*e^3*g)
)/(105*e*(b*e - 2*c*d)^6*(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)*((
(e*(b*e - c*d) + c*d*e)*((16*c^7*e^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^
2 - 4*b*c^2*d*e)) + (32*c^7*e*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2
*d*e)) - (16*b*c^7*e^3*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (8*c^6*e*(15
*b^2*e^2*g + 44*c^2*d^2*g + 16*b*c*e^2*f - 68*c^2*d*e*f - 22*b*c*d*e*g))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2
*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^6*e^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e
^2 - 4*b*c^2*d*e)) - (32*c^7*d*e*g*(b*e - c*d))/(105*(b*e - 2*c*d)^6*(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^7*e^2*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e
^2 - 4*b*c^2*d*e)) + (32*c^7*e*g*(e*(b*e - c*d) + c*d*e))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^
2*d*e)) - (16*b*c^7*e^3*g)/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*(216
*b^2*c^4*e^4*f - 246*b^3*c^3*e^4*g + 1264*c^6*d^2*e^2*f + 592*c^6*d^3*e*g - 992*b*c^5*d*e^3*f - 1696*b*c^5*d^2
*e^2*g + 1140*b^2*c^4*d*e^3*g))/(105*e*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (4*b*c^5*e*(15
*b^2*e^2*g + 44*c^2*d^2*g + 16*b*c*e^2*f - 68*c^2*d*e*f - 22*b*c*d*e*g))/(105*(b*e - 2*c*d)^6*(4*c^3*d^2 + b^2
*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (b*c*(2112*c^6*d^4*g - 538*b^3*c^3*e^4*f + 452*b^4*c^2*e^4*g + 3264*c^6*d^3
*e*f - 6152*b*c^5*d^3*e*g - 5528*b*c^5*d^2*e^2*f + 3012*b^2*c^4*d*e^3*f - 2832*b^3*c^3*d*e^3*g + 6420*b^2*c^4*
d^2*e^2*g))/(105*e*(b*e - 2*c*d)^6*(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 {3}{2}} \left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)**3), x)

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