3.21.38 \(\int \frac {(d+e x)^{9/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=369 \[ \frac {32 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^5 e^2 \sqrt {d+e x}}+\frac {16 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^4 e^2}+\frac {12 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^3 e^2}+\frac {2 (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{7 c^2 e^2 (2 c d-b e)}+\frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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Rubi [A]  time = 0.55, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {788, 656, 648} \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{7 c^2 e^2 (2 c d-b e)}+\frac {12 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^3 e^2}+\frac {16 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^4 e^2}+\frac {32 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^5 e^2 \sqrt {d+e x}}+\frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (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[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(9/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(32*(2*c*d - b*e)^2*(7*c*e*f + 9*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(35*c^5*e^2*Sqrt[
d + e*x]) + (16*(2*c*d - b*e)*(7*c*e*f + 9*c*d*g - 8*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2])/(35*c^4*e^2) + (12*(7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2])/(35*c^3*e^2) + (2*(7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]
)/(7*c^2*e^2*(2*c*d - b*e))

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), 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 + p, 0]

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 788

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(7 c e f+9 c d g-8 b e g) \int \frac {(d+e x)^{7/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac {(6 (7 c e f+9 c d g-8 b e g)) \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{7 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {12 (7 c e f+9 c d g-8 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^3 e^2}+\frac {2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac {(24 (2 c d-b e) (7 c e f+9 c d g-8 b e g)) \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{35 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (2 c d-b e) (7 c e f+9 c d g-8 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^4 e^2}+\frac {12 (7 c e f+9 c d g-8 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^3 e^2}+\frac {2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac {\left (16 (2 c d-b e)^2 (7 c e f+9 c d g-8 b e g)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{35 c^4 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e)^2 (7 c e f+9 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^5 e^2 \sqrt {d+e x}}+\frac {16 (2 c d-b e) (7 c e f+9 c d g-8 b e g) \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^4 e^2}+\frac {12 (7 c e f+9 c d g-8 b e g) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^3 e^2}+\frac {2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 245, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-128 b^4 e^4 g+16 b^3 c e^3 (53 d g+7 e f-4 e g x)-8 b^2 c^2 e^2 \left (257 d^2 g+d e (77 f-45 g x)-e^2 x (7 f+2 g x)\right )-2 b c^3 e \left (-1075 d^3 g+d^2 e (334 g x-553 f)+d e^2 x (126 f+37 g x)+e^3 x^2 (7 f+4 g x)\right )+c^4 \left (-814 d^4 g+d^3 e (407 g x-637 f)+d^2 e^2 x (301 f+93 g x)+d e^3 x^2 (49 f+29 g x)+e^4 x^3 (7 f+5 g x)\right )\right )}{35 c^5 e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-128*b^4*e^4*g + 16*b^3*c*e^3*(7*e*f + 53*d*g - 4*e*g*x) - 8*b^2*c^2*e^2*(257*d^2*g + d*e*(
77*f - 45*g*x) - e^2*x*(7*f + 2*g*x)) - 2*b*c^3*e*(-1075*d^3*g + e^3*x^2*(7*f + 4*g*x) + d*e^2*x*(126*f + 37*g
*x) + d^2*e*(-553*f + 334*g*x)) + c^4*(-814*d^4*g + e^4*x^3*(7*f + 5*g*x) + d*e^3*x^2*(49*f + 29*g*x) + d^2*e^
2*x*(301*f + 93*g*x) + d^3*e*(-637*f + 407*g*x))))/(35*c^5*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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IntegrateAlgebraic [A]  time = 5.58, size = 418, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (-128 b^4 e^4 g-64 b^3 c e^3 g (d+e x)+912 b^3 c d e^3 g+112 b^3 c e^4 f-2400 b^2 c^2 d^2 e^2 g+56 b^2 c^2 e^3 f (d+e x)-672 b^2 c^2 d e^3 f+16 b^2 c^2 e^2 g (d+e x)^2+328 b^2 c^2 d e^2 g (d+e x)+2752 b c^3 d^3 e g+1344 b c^3 d^2 e^2 f-544 b c^3 d^2 e g (d+e x)-14 b c^3 e^2 f (d+e x)^2-224 b c^3 d e^2 f (d+e x)-8 b c^3 e g (d+e x)^3-50 b c^3 d e g (d+e x)^2-1152 c^4 d^4 g-896 c^4 d^3 e f+288 c^4 d^3 g (d+e x)+224 c^4 d^2 e f (d+e x)+36 c^4 d^2 g (d+e x)^2+7 c^4 e f (d+e x)^3+28 c^4 d e f (d+e x)^2+5 c^4 g (d+e x)^4+9 c^4 d g (d+e x)^3\right )}{35 c^5 e^2 \sqrt {d+e x} (b e+c (d+e x)-2 c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2]*(-896*c^4*d^3*e*f + 1344*b*c^3*d^2*e^2*f - 672*b^2*c^2*d*e^3*
f + 112*b^3*c*e^4*f - 1152*c^4*d^4*g + 2752*b*c^3*d^3*e*g - 2400*b^2*c^2*d^2*e^2*g + 912*b^3*c*d*e^3*g - 128*b
^4*e^4*g + 224*c^4*d^2*e*f*(d + e*x) - 224*b*c^3*d*e^2*f*(d + e*x) + 56*b^2*c^2*e^3*f*(d + e*x) + 288*c^4*d^3*
g*(d + e*x) - 544*b*c^3*d^2*e*g*(d + e*x) + 328*b^2*c^2*d*e^2*g*(d + e*x) - 64*b^3*c*e^3*g*(d + e*x) + 28*c^4*
d*e*f*(d + e*x)^2 - 14*b*c^3*e^2*f*(d + e*x)^2 + 36*c^4*d^2*g*(d + e*x)^2 - 50*b*c^3*d*e*g*(d + e*x)^2 + 16*b^
2*c^2*e^2*g*(d + e*x)^2 + 7*c^4*e*f*(d + e*x)^3 + 9*c^4*d*g*(d + e*x)^3 - 8*b*c^3*e*g*(d + e*x)^3 + 5*c^4*g*(d
+ e*x)^4))/(35*c^5*e^2*Sqrt[d + e*x]*(-2*c*d + b*e + c*(d + e*x)))

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fricas [A]  time = 0.41, size = 374, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (5 \, c^{4} e^{4} g x^{4} + {\left (7 \, c^{4} e^{4} f + {\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} + {\left (7 \, {\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f + {\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 7 \, {\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f - 2 \, {\left (407 \, c^{4} d^{4} - 1075 \, b c^{3} d^{3} e + 1028 \, b^{2} c^{2} d^{2} e^{2} - 424 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g + {\left (7 \, {\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f + {\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{35 \, {\left (c^{6} e^{4} x^{2} + b c^{5} e^{4} x - c^{6} d^{2} e^{2} + b c^{5} d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*c^4*e^4*g*x^4 + (7*c^4*e^4*f + (29*c^4*d*e^3 - 8*b*c^3*e^4)*g)*x^3 + (7*(7*c^4*d*e^3 - 2*b*c^3*e^4)*f
+ (93*c^4*d^2*e^2 - 74*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*g)*x^2 - 7*(91*c^4*d^3*e - 158*b*c^3*d^2*e^2 + 88*b^2*c^2
*d*e^3 - 16*b^3*c*e^4)*f - 2*(407*c^4*d^4 - 1075*b*c^3*d^3*e + 1028*b^2*c^2*d^2*e^2 - 424*b^3*c*d*e^3 + 64*b^4
*e^4)*g + (7*(43*c^4*d^2*e^2 - 36*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*f + (407*c^4*d^3*e - 668*b*c^3*d^2*e^2 + 360*b^
2*c^2*d*e^3 - 64*b^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^6*e^4*x^2 + b*c^
5*e^4*x - c^6*d^2*e^2 + b*c^5*d*e^3)

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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((e*x+d)^(9/2)*(g*x+f)/(-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:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [b,c,d,exp(1),exp(2)]=[-80,76,14,-33,-32]Warning, need to choose a branch for the root of a poly
nomial with parameters. This might be wrong.The choice was done assuming [b,c,d,exp(1),exp(2)]=[-69,66,23,-29,
45]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done assuming [b,c,d,exp(1),exp(2)]=[66,44,20,6,-14]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [b,c,d,exp(1),exp(2)]=[87,21,9,-1
4,37]Evaluation time: 119.18Unable to transpose Error: Bad Argument Value

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maple [A]  time = 0.05, size = 367, normalized size = 0.99 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-5 g \,e^{4} x^{4} c^{4}+8 b \,c^{3} e^{4} g \,x^{3}-29 c^{4} d \,e^{3} g \,x^{3}-7 c^{4} e^{4} f \,x^{3}-16 b^{2} c^{2} e^{4} g \,x^{2}+74 b \,c^{3} d \,e^{3} g \,x^{2}+14 b \,c^{3} e^{4} f \,x^{2}-93 c^{4} d^{2} e^{2} g \,x^{2}-49 c^{4} d \,e^{3} f \,x^{2}+64 b^{3} c \,e^{4} g x -360 b^{2} c^{2} d \,e^{3} g x -56 b^{2} c^{2} e^{4} f x +668 b \,c^{3} d^{2} e^{2} g x +252 b \,c^{3} d \,e^{3} f x -407 c^{4} d^{3} e g x -301 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -848 b^{3} c d \,e^{3} g -112 b^{3} c \,e^{4} f +2056 b^{2} c^{2} d^{2} e^{2} g +616 b^{2} c^{2} d \,e^{3} f -2150 b \,c^{3} d^{3} e g -1106 b \,c^{3} d^{2} e^{2} f +814 c^{4} d^{4} g +637 f \,d^{3} c^{4} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{35 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}} c^{5} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)*(g*x+f)/(-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)*(-5*c^4*e^4*g*x^4+8*b*c^3*e^4*g*x^3-29*c^4*d*e^3*g*x^3-7*c^4*e^4*f*x^3-16*b^2*c^2*e^4*g*
x^2+74*b*c^3*d*e^3*g*x^2+14*b*c^3*e^4*f*x^2-93*c^4*d^2*e^2*g*x^2-49*c^4*d*e^3*f*x^2+64*b^3*c*e^4*g*x-360*b^2*c
^2*d*e^3*g*x-56*b^2*c^2*e^4*f*x+668*b*c^3*d^2*e^2*g*x+252*b*c^3*d*e^3*f*x-407*c^4*d^3*e*g*x-301*c^4*d^2*e^2*f*
x+128*b^4*e^4*g-848*b^3*c*d*e^3*g-112*b^3*c*e^4*f+2056*b^2*c^2*d^2*e^2*g+616*b^2*c^2*d*e^3*f-2150*b*c^3*d^3*e*
g-1106*b*c^3*d^2*e^2*f+814*c^4*d^4*g+637*c^4*d^3*e*f)*(e*x+d)^(3/2)/c^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 [A]  time = 0.74, size = 317, normalized size = 0.86 \begin {gather*} -\frac {2 \, {\left (c^{3} e^{3} x^{3} - 91 \, c^{3} d^{3} + 158 \, b c^{2} d^{2} e - 88 \, b^{2} c d e^{2} + 16 \, b^{3} e^{3} + {\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} + {\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{5 \, \sqrt {-c e x + c d - b e} c^{4} e} - \frac {2 \, {\left (5 \, c^{4} e^{4} x^{4} - 814 \, c^{4} d^{4} + 2150 \, b c^{3} d^{3} e - 2056 \, b^{2} c^{2} d^{2} e^{2} + 848 \, b^{3} c d e^{3} - 128 \, b^{4} e^{4} + {\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + {\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{35 \, \sqrt {-c e x + c d - b e} c^{5} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/5*(c^3*e^3*x^3 - 91*c^3*d^3 + 158*b*c^2*d^2*e - 88*b^2*c*d*e^2 + 16*b^3*e^3 + (7*c^3*d*e^2 - 2*b*c^2*e^3)*x
^2 + (43*c^3*d^2*e - 36*b*c^2*d*e^2 + 8*b^2*c*e^3)*x)*f/(sqrt(-c*e*x + c*d - b*e)*c^4*e) - 2/35*(5*c^4*e^4*x^4
- 814*c^4*d^4 + 2150*b*c^3*d^3*e - 2056*b^2*c^2*d^2*e^2 + 848*b^3*c*d*e^3 - 128*b^4*e^4 + (29*c^4*d*e^3 - 8*b
*c^3*e^4)*x^3 + (93*c^4*d^2*e^2 - 74*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*x^2 + (407*c^4*d^3*e - 668*b*c^3*d^2*e^2 +
360*b^2*c^2*d*e^3 - 64*b^3*c*e^4)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^5*e^2)

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mupad [B]  time = 3.09, size = 398, normalized size = 1.08 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,g\,x^4\,\sqrt {d+e\,x}}{7\,c^2}-\frac {\sqrt {d+e\,x}\,\left (256\,g\,b^4\,e^4-1696\,g\,b^3\,c\,d\,e^3-224\,f\,b^3\,c\,e^4+4112\,g\,b^2\,c^2\,d^2\,e^2+1232\,f\,b^2\,c^2\,d\,e^3-4300\,g\,b\,c^3\,d^3\,e-2212\,f\,b\,c^3\,d^2\,e^2+1628\,g\,c^4\,d^4+1274\,f\,c^4\,d^3\,e\right )}{35\,c^6\,e^4}+\frac {x^2\,\sqrt {d+e\,x}\,\left (32\,g\,b^2\,c^2\,e^4-148\,g\,b\,c^3\,d\,e^3-28\,f\,b\,c^3\,e^4+186\,g\,c^4\,d^2\,e^2+98\,f\,c^4\,d\,e^3\right )}{35\,c^6\,e^4}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (29\,c\,d\,g-8\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^3\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,g\,b^3\,c\,e^4+720\,g\,b^2\,c^2\,d\,e^3+112\,f\,b^2\,c^2\,e^4-1336\,g\,b\,c^3\,d^2\,e^2-504\,f\,b\,c^3\,d\,e^3+814\,g\,c^4\,d^3\,e+602\,f\,c^4\,d^2\,e^2\right )}{35\,c^6\,e^4}\right )}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(d + e*x)^(9/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2),x)

[Out]

((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*g*x^4*(d + e*x)^(1/2))/(7*c^2) - ((d + e*x)^(1/2)*(256*b^4*e^
4*g + 1628*c^4*d^4*g - 224*b^3*c*e^4*f + 1274*c^4*d^3*e*f - 4300*b*c^3*d^3*e*g - 1696*b^3*c*d*e^3*g - 2212*b*c
^3*d^2*e^2*f + 1232*b^2*c^2*d*e^3*f + 4112*b^2*c^2*d^2*e^2*g))/(35*c^6*e^4) + (x^2*(d + e*x)^(1/2)*(32*b^2*c^2
*e^4*g + 186*c^4*d^2*e^2*g - 28*b*c^3*e^4*f + 98*c^4*d*e^3*f - 148*b*c^3*d*e^3*g))/(35*c^6*e^4) + (2*x^3*(d +
e*x)^(1/2)*(29*c*d*g - 8*b*e*g + 7*c*e*f))/(35*c^3*e) + (x*(d + e*x)^(1/2)*(112*b^2*c^2*e^4*f + 602*c^4*d^2*e^
2*f - 128*b^3*c*e^4*g + 814*c^4*d^3*e*g - 504*b*c^3*d*e^3*f - 1336*b*c^3*d^2*e^2*g + 720*b^2*c^2*d*e^3*g))/(35
*c^6*e^4)))/(x^2 + (b*x)/c + (d*(b*e - c*d))/(c*e^2))

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sympy [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((e*x+d)**(9/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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