3.21.48 \(\int \frac {(d+e x)^{9/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=291 \[ -\frac {16 \sqrt {d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{5/2} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

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Rubi [A]  time = 0.41, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 4, 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} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {16 \sqrt {d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \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[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.15, size = 180, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-16 b^3 e^3 g+8 b^2 c e^2 (8 d g+e (f-3 g x))-2 b c^2 e \left (41 d^2 g+2 d e (5 f-18 g x)+3 e^2 x (g x-2 f)\right )+c^3 \left (34 d^3 g+d^2 e (11 f-51 g x)+6 d e^2 x (2 g x-3 f)+e^3 x^2 (3 f+g x)\right )\right )}{3 c^4 e^2 (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[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(-16*b^3*e^3*g + 8*b^2*c*e^2*(8*d*g + e*(f - 3*g*x)) - 2*b*c^2*e*(41*d^2*g + 2*d*e*(5*f - 18*
g*x) + 3*e^2*x*(-2*f + g*x)) + c^3*(34*d^3*g + d^2*e*(11*f - 51*g*x) + e^3*x^2*(3*f + g*x) + 6*d*e^2*x*(-3*f +
2*g*x))))/(3*c^4*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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IntegrateAlgebraic [A]  time = 5.96, size = 247, normalized size = 0.85 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (-16 b^3 e^3 g-24 b^2 c e^2 g (d+e x)+88 b^2 c d e^2 g+8 b^2 c e^3 f-160 b c^2 d^2 e g+12 b c^2 e^2 f (d+e x)-32 b c^2 d e^2 f-6 b c^2 e g (d+e x)^2+84 b c^2 d e g (d+e x)+96 c^3 d^3 g+32 c^3 d^2 e f-72 c^3 d^2 g (d+e x)+3 c^3 e f (d+e x)^2-24 c^3 d e f (d+e x)+c^3 g (d+e x)^3+9 c^3 d g (d+e x)^2\right )}{3 c^4 e^2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2}} \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)^(5/2),x]

[Out]

(-2*(d + e*x)^(3/2)*(32*c^3*d^2*e*f - 32*b*c^2*d*e^2*f + 8*b^2*c*e^3*f + 96*c^3*d^3*g - 160*b*c^2*d^2*e*g + 88
*b^2*c*d*e^2*g - 16*b^3*e^3*g - 24*c^3*d*e*f*(d + e*x) + 12*b*c^2*e^2*f*(d + e*x) - 72*c^3*d^2*g*(d + e*x) + 8
4*b*c^2*d*e*g*(d + e*x) - 24*b^2*c*e^2*g*(d + e*x) + 3*c^3*e*f*(d + e*x)^2 + 9*c^3*d*g*(d + e*x)^2 - 6*b*c^2*e
*g*(d + e*x)^2 + c^3*g*(d + e*x)^3))/(3*c^4*e^2*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(3/2))

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fricas [A]  time = 0.41, size = 308, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (c^{3} e^{3} g x^{3} + 3 \, {\left (c^{3} e^{3} f + 2 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} + {\left (11 \, c^{3} d^{2} e - 20 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \, {\left (17 \, c^{3} d^{3} - 41 \, b c^{2} d^{2} e + 32 \, b^{2} c d e^{2} - 8 \, b^{3} e^{3}\right )} g - 3 \, {\left (2 \, {\left (3 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\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}}{3 \, {\left (c^{6} e^{5} x^{3} + c^{6} d^{3} e^{2} - 2 \, b c^{5} d^{2} e^{3} + b^{2} c^{4} d e^{4} - {\left (c^{6} d e^{4} - 2 \, b c^{5} e^{5}\right )} x^{2} - {\left (c^{6} d^{2} e^{3} - b^{2} c^{4} e^{5}\right )} x\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)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(c^3*e^3*g*x^3 + 3*(c^3*e^3*f + 2*(2*c^3*d*e^2 - b*c^2*e^3)*g)*x^2 + (11*c^3*d^2*e - 20*b*c^2*d*e^2 + 8*b
^2*c*e^3)*f + 2*(17*c^3*d^3 - 41*b*c^2*d^2*e + 32*b^2*c*d*e^2 - 8*b^3*e^3)*g - 3*(2*(3*c^3*d*e^2 - 2*b*c^2*e^3
)*f + (17*c^3*d^2*e - 24*b*c^2*d*e^2 + 8*b^2*c*e^3)*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^5*x^3 + c^6*d^3*e^2 - 2*b*c^5*d^2*e^3 + b^2*c^4*d*e^4 - (c^6*d*e^4 - 2*b*c^5*e^5)*x^2 - (c^6*d^2*e
^3 - b^2*c^4*e^5)*x)

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giac [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)^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 235, normalized size = 0.81 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-g \,e^{3} x^{3} c^{3}+6 b \,c^{2} e^{3} g \,x^{2}-12 c^{3} d \,e^{2} g \,x^{2}-3 c^{3} e^{3} f \,x^{2}+24 b^{2} c \,e^{3} g x -72 b \,c^{2} d \,e^{2} g x -12 b \,c^{2} e^{3} f x +51 c^{3} d^{2} e g x +18 c^{3} d \,e^{2} f x +16 b^{3} e^{3} g -64 b^{2} c d \,e^{2} g -8 b^{2} c \,e^{3} f +82 b \,c^{2} d^{2} e g +20 b \,c^{2} d \,e^{2} f -34 c^{3} d^{3} g -11 f \,d^{2} c^{3} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} c^{4} 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)^(5/2),x)

[Out]

-2/3*(c*e*x+b*e-c*d)*(-c^3*e^3*g*x^3+6*b*c^2*e^3*g*x^2-12*c^3*d*e^2*g*x^2-3*c^3*e^3*f*x^2+24*b^2*c*e^3*g*x-72*
b*c^2*d*e^2*g*x-12*b*c^2*e^3*f*x+51*c^3*d^2*e*g*x+18*c^3*d*e^2*f*x+16*b^3*e^3*g-64*b^2*c*d*e^2*g-8*b^2*c*e^3*f
+82*b*c^2*d^2*e*g+20*b*c^2*d*e^2*f-34*c^3*d^3*g-11*c^3*d^2*e*f)*(e*x+d)^(5/2)/c^4/e^2/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(5/2)

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maxima [A]  time = 0.96, size = 246, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} + 11 \, c^{2} d^{2} - 20 \, b c d e + 8 \, b^{2} e^{2} - 6 \, {\left (3 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, {\left (c^{4} e^{2} x - c^{4} d e + b c^{3} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (c^{3} e^{3} x^{3} + 34 \, c^{3} d^{3} - 82 \, b c^{2} d^{2} e + 64 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 6 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} g}{3 \, {\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )} \sqrt {-c e x + c d - b e}} \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)^(5/2),x, algorithm="maxima")

[Out]

2/3*(3*c^2*e^2*x^2 + 11*c^2*d^2 - 20*b*c*d*e + 8*b^2*e^2 - 6*(3*c^2*d*e - 2*b*c*e^2)*x)*f/((c^4*e^2*x - c^4*d*
e + b*c^3*e^2)*sqrt(-c*e*x + c*d - b*e)) + 2/3*(c^3*e^3*x^3 + 34*c^3*d^3 - 82*b*c^2*d^2*e + 64*b^2*c*d*e^2 - 1
6*b^3*e^3 + 6*(2*c^3*d*e^2 - b*c^2*e^3)*x^2 - 3*(17*c^3*d^2*e - 24*b*c^2*d*e^2 + 8*b^2*c*e^3)*x)*g/((c^5*e^3*x
- c^5*d*e^2 + b*c^4*e^3)*sqrt(-c*e*x + c*d - b*e))

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mupad [B]  time = 3.21, size = 314, 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 {\sqrt {d+e\,x}\,\left (-32\,g\,b^3\,e^3+128\,g\,b^2\,c\,d\,e^2+16\,f\,b^2\,c\,e^3-164\,g\,b\,c^2\,d^2\,e-40\,f\,b\,c^2\,d\,e^2+68\,g\,c^3\,d^3+22\,f\,c^3\,d^2\,e\right )}{3\,c^6\,e^5}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (4\,c\,d\,g-2\,b\,e\,g+c\,e\,f\right )}{c^4\,e^3}+\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{3\,c^3\,e^2}-\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-144\,g\,b\,c^2\,d\,e^2-24\,f\,b\,c^2\,e^3+102\,g\,c^3\,d^2\,e+36\,f\,c^3\,d\,e^2\right )}{3\,c^6\,e^5}\right )}{x^3+\frac {x\,\left (3\,b^2\,c^4\,e^5-3\,c^6\,d^2\,e^3\right )}{3\,c^6\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \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)^(5/2),x)

[Out]

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(((d + e*x)^(1/2)*(68*c^3*d^3*g - 32*b^3*e^3*g + 16*b^2*c*e^3*f
+ 22*c^3*d^2*e*f - 40*b*c^2*d*e^2*f - 164*b*c^2*d^2*e*g + 128*b^2*c*d*e^2*g))/(3*c^6*e^5) + (2*x^2*(d + e*x)^(
1/2)*(4*c*d*g - 2*b*e*g + c*e*f))/(c^4*e^3) + (2*g*x^3*(d + e*x)^(1/2))/(3*c^3*e^2) - (x*(d + e*x)^(1/2)*(48*b
^2*c*e^3*g - 24*b*c^2*e^3*f + 36*c^3*d*e^2*f + 102*c^3*d^2*e*g - 144*b*c^2*d*e^2*g))/(3*c^6*e^5)))/(x^3 + (x*(
3*b^2*c^4*e^5 - 3*c^6*d^2*e^3))/(3*c^6*e^5) + (d*(b*e - c*d)^2)/(c^2*e^3) + (x^2*(2*b*e - c*d))/(c*e))

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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)**(5/2),x)

[Out]

Timed out

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