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

Optimal. Leaf size=136 \[ \frac {2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{3 e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{3 e^2 (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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Rubi [A]  time = 0.14, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {792, 613} \begin {gather*} \frac {2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{3 e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{3 e^2 (d+e x) (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)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

(2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(3*e*(2*c*d - b*e)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
- (2*(e*f - d*g))/(3*e^2*(2*c*d - b*e)*(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 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 )^{3/2}} \, dx &=-\frac {2 (e f-d g)}{3 e^2 (2 c d-b e) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(4 c e f+2 c d g-3 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}{3 e (2 c d-b e)}\\ &=\frac {2 (4 c e f+2 c d g-3 b e g) (b+2 c x)}{3 e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{3 e^2 (2 c d-b e) (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 = 0.10, size = 149, normalized size = 1.10 \begin {gather*} \frac {2 b^2 e^2 (2 d g+e (f+3 g x))+4 b c e \left (d^2 g+d e (2 g x-4 f)+e^2 x (3 g x-2 f)\right )-8 c^2 \left (d^3 g+d^2 e (g x-f)+d e^2 x (2 f+g x)+2 e^3 f x^2\right )}{3 e^2 (d+e x) (b e-2 c d)^3 \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)^(3/2)),x]

[Out]

(-8*c^2*(d^3*g + 2*e^3*f*x^2 + d^2*e*(-f + g*x) + d*e^2*x*(2*f + g*x)) + 4*b*c*e*(d^2*g + d*e*(-4*f + 2*g*x) +
e^2*x*(-2*f + 3*g*x)) + 2*b^2*e^2*(2*d*g + e*(f + 3*g*x)))/(3*e^2*(-2*c*d + b*e)^3*(d + e*x)*Sqrt[(d + e*x)*(
-(b*e) + c*(d - e*x))])

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IntegrateAlgebraic [B]  time = 127.73, size = 20130, normalized size = 148.01 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [B]  time = 7.16, size = 407, normalized size = 2.99 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} e^{3} f + {\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} g\right )} x^{2} - {\left (4 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} f + 2 \, {\left (2 \, c^{2} d^{3} - b c d^{2} e - b^{2} d e^{2}\right )} g + {\left (4 \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} f + {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} - 3 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, {\left (8 \, c^{4} d^{6} e^{2} - 20 \, b c^{3} d^{5} e^{3} + 18 \, b^{2} c^{2} d^{4} e^{4} - 7 \, b^{3} c d^{3} e^{5} + b^{4} d^{2} e^{6} - {\left (8 \, c^{4} d^{3} e^{5} - 12 \, b c^{3} d^{2} e^{6} + 6 \, b^{2} c^{2} d e^{7} - b^{3} c e^{8}\right )} x^{3} - {\left (8 \, c^{4} d^{4} e^{4} - 4 \, b c^{3} d^{3} e^{5} - 6 \, b^{2} c^{2} d^{2} e^{6} + 5 \, b^{3} c d e^{7} - b^{4} e^{8}\right )} x^{2} + {\left (8 \, c^{4} d^{5} e^{3} - 28 \, b c^{3} d^{4} e^{4} + 30 \, b^{2} c^{2} d^{3} e^{5} - 13 \, b^{3} c d^{2} e^{6} + 2 \, b^{4} d e^{7}\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)^(3/2),x, algorithm="fricas")

[Out]

2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f + (2*c^2*d*e^2 - 3*b*c*e^3)*g)*x^2 - (4*c^2*d^2
*e - 8*b*c*d*e^2 + b^2*e^3)*f + 2*(2*c^2*d^3 - b*c*d^2*e - b^2*d*e^2)*g + (4*(2*c^2*d*e^2 + b*c*e^3)*f + (4*c^
2*d^2*e - 4*b*c*d*e^2 - 3*b^2*e^3)*g)*x)/(8*c^4*d^6*e^2 - 20*b*c^3*d^5*e^3 + 18*b^2*c^2*d^4*e^4 - 7*b^3*c*d^3*
e^5 + b^4*d^2*e^6 - (8*c^4*d^3*e^5 - 12*b*c^3*d^2*e^6 + 6*b^2*c^2*d*e^7 - b^3*c*e^8)*x^3 - (8*c^4*d^4*e^4 - 4*
b*c^3*d^3*e^5 - 6*b^2*c^2*d^2*e^6 + 5*b^3*c*d*e^7 - b^4*e^8)*x^2 + (8*c^4*d^5*e^3 - 28*b*c^3*d^4*e^4 + 30*b^2*
c^2*d^3*e^5 - 13*b^3*c*d^2*e^6 + 2*b^4*d*e^7)*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)^(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 [A]  time = 0.05, size = 228, normalized size = 1.68 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (6 b c \,e^{3} g \,x^{2}-4 c^{2} d \,e^{2} g \,x^{2}-8 c^{2} e^{3} f \,x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -4 b c \,e^{3} f x -4 c^{2} d^{2} e g x -8 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +b^{2} e^{3} f +2 b c \,d^{2} e g -8 b c d \,e^{2} f -4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right )}{3 \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\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)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/3*(c*e*x+b*e-c*d)*(6*b*c*e^3*g*x^2-4*c^2*d*e^2*g*x^2-8*c^2*e^3*f*x^2+3*b^2*e^3*g*x+4*b*c*d*e^2*g*x-4*b*c*e^
3*f*x-4*c^2*d^2*e*g*x-8*c^2*d*e^2*f*x+2*b^2*d*e^2*g+b^2*e^3*f+2*b*c*d^2*e*g-8*b*c*d*e^2*f-4*c^2*d^3*g+4*c^2*d^
2*e*f)/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/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)/(-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 = 3.20, size = 872, normalized size = 6.41 \begin {gather*} \frac {\left (\frac {2\,b\,g}{3\,e\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {4\,c\,d\,g}{3\,e^2\,{\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}}{d+e\,x}+\frac {\left (\frac {2\,d\,g}{3\,b^2\,e^4-12\,b\,c\,d\,e^3+12\,c^2\,d^2\,e^2}-\frac {2\,e\,f}{3\,b^2\,e^4-12\,b\,c\,d\,e^3+12\,c^2\,d^2\,e^2}\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 (x\,\left (\frac {\left (e\,\left (b\,e-c\,d\right )+c\,d\,e\right )\,\left (\frac {4\,c^3\,e\,\left (3\,b\,g-2\,c\,f\right )}{3\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}+\frac {4\,b\,c^3\,e\,g}{3\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}-\frac {8\,c^3\,g\,\left (e\,\left (b\,e-c\,d\right )+c\,d\,e\right )}{3\,e\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}\right )}{c\,e^2}-\frac {2\,c^2\,\left (8\,g\,b^2\,e^2-16\,g\,b\,c\,d\,e-10\,f\,b\,c\,e^2+8\,g\,c^2\,d^2+16\,f\,c^2\,d\,e\right )}{3\,e\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}-\frac {2\,b\,c^2\,e\,\left (3\,b\,g-2\,c\,f\right )}{3\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}+\frac {8\,c^3\,d\,g\,\left (b\,e-c\,d\right )}{3\,e\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}\right )+\frac {d\,\left (b\,e-c\,d\right )\,\left (\frac {4\,c^3\,e\,\left (3\,b\,g-2\,c\,f\right )}{3\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}+\frac {4\,b\,c^3\,e\,g}{3\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}-\frac {8\,c^3\,g\,\left (e\,\left (b\,e-c\,d\right )+c\,d\,e\right )}{3\,e\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}\right )}{c\,e^2}-\frac {b\,c\,\left (8\,g\,b^2\,e^2-16\,g\,b\,c\,d\,e-10\,f\,b\,c\,e^2+8\,g\,c^2\,d^2+16\,f\,c^2\,d\,e\right )}{3\,e\,{\left (b\,e-2\,c\,d\right )}^2\,\left (b^2\,c\,e^2-4\,b\,c^2\,d\,e+4\,c^3\,d^2\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )} \end {gather*}

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)^(3/2)),x)

[Out]

(((2*b*g)/(3*e*(b*e - 2*c*d)^3) - (4*c*d*g)/(3*e^2*(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*g)/(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3) - (2*e*f)/(3*b^2*e^4 + 12*c^2*d^2*e^2 -
12*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)^2 + ((x*(((e*(b*e - c*d) + c*d*e)*((4*c^
3*e*(3*b*g - 2*c*f))/(3*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (4*b*c^3*e*g)/(3*(b*e - 2*c*d
)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^3*g*(e*(b*e - c*d) + c*d*e))/(3*e*(b*e - 2*c*d)^2*(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*e^2*g + 8*c^2*d^2*g - 10*b*c*e^2*f + 16*c^2*d*e*f - 16*
b*c*d*e*g))/(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (2*b*c^2*e*(3*b*g - 2*c*f))/(3*(b*e
- 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (8*c^3*d*g*(b*e - c*d))/(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 +
b^2*c*e^2 - 4*b*c^2*d*e))) + (d*(b*e - c*d)*((4*c^3*e*(3*b*g - 2*c*f))/(3*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e
^2 - 4*b*c^2*d*e)) + (4*b*c^3*e*g)/(3*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^3*g*(e*(b*
e - c*d) + c*d*e))/(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - (b*c*(8*b^2*e^2*g +
8*c^2*d^2*g - 10*b*c*e^2*f + 16*c^2*d*e*f - 16*b*c*d*e*g))/(3*e*(b*e - 2*c*d)^2*(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 )}\, 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)**(3/2),x)

[Out]

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

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