3.20.86 \(\int \frac {f+g x}{(d+e x)^5 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=360 \[ -\frac {32 c^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x) (2 c d-b e)^5}-\frac {16 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x)^2 (2 c d-b e)^4}-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{63 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (d+e x)^5 (2 c d-b e)} \]

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Rubi [A]  time = 0.56, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {792, 658, 650} \begin {gather*} -\frac {32 c^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x) (2 c d-b e)^5}-\frac {16 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x)^2 (2 c d-b e)^4}-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{63 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (d+e x)^5 (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(9*e^2*(2*c*d - b*e)*(d + e*x)^5) - (2*(8*c*e*f + 1
0*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^4) - (4*c*(8*c
*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^3) -
(16*c^2*(8*c*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(315*e^2*(2*c*d - b*e)^4*(d
+ e*x)^2) - (32*c^3*(8*c*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(315*e^2*(2*c*d
- b*e)^5*(d + e*x))

Rule 650

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))/((p + 1)*(2*c*d - b*e)), 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] && EqQ[m + 2*p + 2, 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)^5 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}+\frac {(8 c e f+10 c d g-9 b e g) \int \frac {1}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{9 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}+\frac {(2 c (8 c e f+10 c d g-9 b e g)) \int \frac {1}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{21 e (2 c d-b e)^2}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}+\frac {\left (8 c^2 (8 c e f+10 c d g-9 b e g)\right ) \int \frac {1}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{105 e (2 c d-b e)^3}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}-\frac {16 c^2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^4 (d+e x)^2}+\frac {\left (16 c^3 (8 c e f+10 c d g-9 b e g)\right ) \int \frac {1}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{315 e (2 c d-b e)^4}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}-\frac {16 c^2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^4 (d+e x)^2}-\frac {32 c^3 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^5 (d+e x)}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 348, normalized size = 0.97 \begin {gather*} -\frac {2 (b e-c d+c e x) \left (5 b^4 e^4 (2 d g+7 e f+9 e g x)-2 b^3 c e^3 \left (47 d^2 g+2 d e (80 f+107 g x)+e^2 x (20 f+27 g x)\right )+12 b^2 c^2 e^2 \left (29 d^3 g+2 d^2 e (47 f+67 g x)+d e^2 x (28 f+41 g x)+2 e^3 x^2 (2 f+3 g x)\right )-8 b c^3 e \left (83 d^4 g+d^3 e (232 f+390 g x)+3 d^2 e^2 x (44 f+83 g x)+4 d e^3 x^2 (12 f+25 g x)+2 e^4 x^3 (4 f+9 g x)\right )+16 c^4 \left (25 d^5 g+d^4 e (83 f+125 g x)+5 d^3 e^2 x (20 f+21 g x)+2 d^2 e^3 x^2 (42 f+25 g x)+10 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{315 e^2 (d+e x)^4 (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)^5*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)*(5*b^4*e^4*(7*e*f + 2*d*g + 9*e*g*x) + 12*b^2*c^2*e^2*(29*d^3*g + 2*e^3*x^2*(2*f +
3*g*x) + d*e^2*x*(28*f + 41*g*x) + 2*d^2*e*(47*f + 67*g*x)) - 2*b^3*c*e^3*(47*d^2*g + e^2*x*(20*f + 27*g*x) +
2*d*e*(80*f + 107*g*x)) + 16*c^4*(25*d^5*g + 8*e^5*f*x^4 + 10*d*e^4*x^3*(4*f + g*x) + 5*d^3*e^2*x*(20*f + 21*g
*x) + 2*d^2*e^3*x^2*(42*f + 25*g*x) + d^4*e*(83*f + 125*g*x)) - 8*b*c^3*e*(83*d^4*g + 2*e^4*x^3*(4*f + 9*g*x)
+ 4*d*e^3*x^2*(12*f + 25*g*x) + 3*d^2*e^2*x*(44*f + 83*g*x) + d^3*e*(232*f + 390*g*x))))/(315*e^2*(-2*c*d + b*
e)^5*(d + e*x)^4*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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IntegrateAlgebraic [B]  time = 111.18, size = 13927, normalized size = 38.69 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.05, size = 564, normalized size = 1.57 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-144 b \,c^{3} e^{5} g \,x^{4}+160 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}+72 b^{2} c^{2} e^{5} g \,x^{3}-800 b \,c^{3} d \,e^{4} g \,x^{3}-64 b \,c^{3} e^{5} f \,x^{3}+800 c^{4} d^{2} e^{3} g \,x^{3}+640 c^{4} d \,e^{4} f \,x^{3}-54 b^{3} c \,e^{5} g \,x^{2}+492 b^{2} c^{2} d \,e^{4} g \,x^{2}+48 b^{2} c^{2} e^{5} f \,x^{2}-1992 b \,c^{3} d^{2} e^{3} g \,x^{2}-384 b \,c^{3} d \,e^{4} f \,x^{2}+1680 c^{4} d^{3} e^{2} g \,x^{2}+1344 c^{4} d^{2} e^{3} f \,x^{2}+45 b^{4} e^{5} g x -428 b^{3} c d \,e^{4} g x -40 b^{3} c \,e^{5} f x +1608 b^{2} c^{2} d^{2} e^{3} g x +336 b^{2} c^{2} d \,e^{4} f x -3120 b \,c^{3} d^{3} e^{2} g x -1056 b \,c^{3} d^{2} e^{3} f x +2000 c^{4} d^{4} e g x +1600 c^{4} d^{3} e^{2} f x +10 b^{4} d \,e^{4} g +35 b^{4} e^{5} f -94 b^{3} c \,d^{2} e^{3} g -320 b^{3} c d \,e^{4} f +348 b^{2} c^{2} d^{3} e^{2} g +1128 b^{2} c^{2} d^{2} e^{3} f -664 b \,c^{3} d^{4} e g -1856 b \,c^{3} d^{3} e^{2} f +400 c^{4} d^{5} g +1328 c^{4} d^{4} e f \right )}{315 \left (e x +d \right )^{4} \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 ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(c*e*x+b*e-c*d)*(-144*b*c^3*e^5*g*x^4+160*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4+72*b^2*c^2*e^5*g*x^3-800*b*
c^3*d*e^4*g*x^3-64*b*c^3*e^5*f*x^3+800*c^4*d^2*e^3*g*x^3+640*c^4*d*e^4*f*x^3-54*b^3*c*e^5*g*x^2+492*b^2*c^2*d*
e^4*g*x^2+48*b^2*c^2*e^5*f*x^2-1992*b*c^3*d^2*e^3*g*x^2-384*b*c^3*d*e^4*f*x^2+1680*c^4*d^3*e^2*g*x^2+1344*c^4*
d^2*e^3*f*x^2+45*b^4*e^5*g*x-428*b^3*c*d*e^4*g*x-40*b^3*c*e^5*f*x+1608*b^2*c^2*d^2*e^3*g*x+336*b^2*c^2*d*e^4*f
*x-3120*b*c^3*d^3*e^2*g*x-1056*b*c^3*d^2*e^3*f*x+2000*c^4*d^4*e*g*x+1600*c^4*d^3*e^2*f*x+10*b^4*d*e^4*g+35*b^4
*e^5*f-94*b^3*c*d^2*e^3*g-320*b^3*c*d*e^4*f+348*b^2*c^2*d^3*e^2*g+1128*b^2*c^2*d^2*e^3*f-664*b*c^3*d^4*e*g-185
6*b*c^3*d^3*e^2*f+400*c^4*d^5*g+1328*c^4*d^4*e*f)/(e*x+d)^4/e^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)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)

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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)^5/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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 = 8.01, size = 949, normalized size = 2.64 \begin {gather*} \frac {\left (\frac {88\,c^2\,d\,g+96\,c^2\,e\,f-88\,b\,c\,e\,g}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {32\,c^3\,g\,\left (4\,b\,e-7\,c\,d\right )}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}-\frac {32\,c^4\,d\,g}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}-\frac {\left (\frac {4\,c\,g\,\left (5\,b\,e-8\,c\,d\right )}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {400\,c^3\,d\,g+384\,c^3\,e\,f-400\,b\,c^2\,e\,g}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}+\frac {16\,c^3\,d\,g}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {2\,b\,g}{9\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c\,d\,g}{9\,e\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^4}-\frac {\left (\frac {20\,c\,d\,g-20\,b\,e\,g+16\,c\,e\,f}{9\,e\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}+\frac {4\,c\,d\,g}{9\,e\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^4}+\frac {\left (\frac {2\,f}{9\,b\,e^2-18\,c\,d\,e}-\frac {2\,d\,g}{e\,\left (9\,b\,e^2-18\,c\,d\,e\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^5}+\frac {\left (\frac {16\,c^3\,d\,g}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}+\frac {16\,c^2\,g\,\left (2\,b\,e-5\,c\,d\right )}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {736\,c^4\,d\,g+768\,c^4\,e\,f-736\,b\,c^3\,e\,g}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}-\frac {32\,c^4\,d\,g}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((88*c^2*d*g + 96*c^2*e*f - 88*b*c*e*g)/(63*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^2*d*g)/(63*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 - (((32*c^3*g*(4*b*
e - 7*c*d))/(945*e^2*(b*e - 2*c*d)^5) - (32*c^4*d*g)/(945*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) - (((4*c*g*(5*b*e - 8*c*d))/(63*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^2*d*g)
/(63*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 - (((40
0*c^3*d*g + 384*c^3*e*f - 400*b*c^2*e*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) + (16*c^3*d*g)/(315*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 - (((2*b*g)/(9*(7*
b*e^2 - 14*c*d*e)*(b*e - 2*c*d)) - (4*c*d*g)/(9*e*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*
d*e - b*e^2*x)^(1/2))/(d + e*x)^4 - (((20*c*d*g - 20*b*e*g + 16*c*e*f)/(9*e*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)
) + (4*c*d*g)/(9*e*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)
^4 + (((2*f)/(9*b*e^2 - 18*c*d*e) - (2*d*g)/(e*(9*b*e^2 - 18*c*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1
/2))/(d + e*x)^5 + (((16*c^3*d*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) + (16*c^2*g*(2*b*e - 5*c*d))/(31
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 + (((736*c^
4*d*g + 768*c^4*e*f - 736*b*c^3*e*g)/(945*e^2*(b*e - 2*c*d)^5) - (32*c^4*d*g)/(945*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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