3.979 \(\int \frac{d+e x}{(a+b x+c x^2)^{9/2}} \, dx\)

Optimal. Leaf size=181 \[ \frac{1024 c^2 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^4 \sqrt{a+b x+c x^2}}-\frac{128 c (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{24 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}} \]

[Out]

(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(7*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(7/2)) + (24*(2*c*d - b*e)*(b + 2*c*x)
)/(35*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(5/2)) - (128*c*(2*c*d - b*e)*(b + 2*c*x))/(35*(b^2 - 4*a*c)^3*(a + b*
x + c*x^2)^(3/2)) + (1024*c^2*(2*c*d - b*e)*(b + 2*c*x))/(35*(b^2 - 4*a*c)^4*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.0589463, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {638, 614, 613} \[ \frac{1024 c^2 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^4 \sqrt{a+b x+c x^2}}-\frac{128 c (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{24 (b+2 c x) (2 c d-b e)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + b*x + c*x^2)^(9/2),x]

[Out]

(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(7*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(7/2)) + (24*(2*c*d - b*e)*(b + 2*c*x)
)/(35*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(5/2)) - (128*c*(2*c*d - b*e)*(b + 2*c*x))/(35*(b^2 - 4*a*c)^3*(a + b*
x + c*x^2)^(3/2)) + (1024*c^2*(2*c*d - b*e)*(b + 2*c*x))/(35*(b^2 - 4*a*c)^4*Sqrt[a + b*x + c*x^2])

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (a+b x+c x^2\right )^{9/2}} \, dx &=-\frac{2 (b d-2 a e+(2 c d-b e) x)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}-\frac{(12 (2 c d-b e)) \int \frac{1}{\left (a+b x+c x^2\right )^{7/2}} \, dx}{7 \left (b^2-4 a c\right )}\\ &=-\frac{2 (b d-2 a e+(2 c d-b e) x)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}+\frac{24 (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}+\frac{(192 c (2 c d-b e)) \int \frac{1}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{35 \left (b^2-4 a c\right )^2}\\ &=-\frac{2 (b d-2 a e+(2 c d-b e) x)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}+\frac{24 (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac{128 c (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}-\frac{\left (512 c^2 (2 c d-b e)\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{35 \left (b^2-4 a c\right )^3}\\ &=-\frac{2 (b d-2 a e+(2 c d-b e) x)}{7 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}+\frac{24 (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac{128 c (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac{1024 c^2 (2 c d-b e) (b+2 c x)}{35 \left (b^2-4 a c\right )^4 \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.194159, size = 159, normalized size = 0.88 \[ \frac{2 \left (5 \left (b^2-4 a c\right )^3 (2 a e-b d+b e x-2 c d x)-12 \left (b^2-4 a c\right )^2 (b+2 c x) (a+x (b+c x)) (b e-2 c d)+64 c \left (b^2-4 a c\right ) (b+2 c x) (a+x (b+c x))^2 (b e-2 c d)-512 c^2 (b+2 c x) (a+x (b+c x))^3 (b e-2 c d)\right )}{35 \left (b^2-4 a c\right )^4 (a+x (b+c x))^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a + b*x + c*x^2)^(9/2),x]

[Out]

(2*(5*(b^2 - 4*a*c)^3*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x) - 12*(b^2 - 4*a*c)^2*(-2*c*d + b*e)*(b + 2*c*x)*(a +
x*(b + c*x)) + 64*c*(b^2 - 4*a*c)*(-2*c*d + b*e)*(b + 2*c*x)*(a + x*(b + c*x))^2 - 512*c^2*(-2*c*d + b*e)*(b +
2*c*x)*(a + x*(b + c*x))^3))/(35*(b^2 - 4*a*c)^4*(a + x*(b + c*x))^(7/2))

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Maple [B]  time = 0.01, size = 500, normalized size = 2.8 \begin{align*} -{\frac{2048\,b{c}^{6}e{x}^{7}-4096\,{c}^{7}d{x}^{7}+7168\,{b}^{2}{c}^{5}e{x}^{6}-14336\,b{c}^{6}d{x}^{6}+7168\,ab{c}^{5}e{x}^{5}-14336\,a{c}^{6}d{x}^{5}+8960\,{b}^{3}{c}^{4}e{x}^{5}-17920\,{b}^{2}{c}^{5}d{x}^{5}+17920\,a{b}^{2}{c}^{4}e{x}^{4}-35840\,ab{c}^{5}d{x}^{4}+4480\,{b}^{4}{c}^{3}e{x}^{4}-8960\,{b}^{3}{c}^{4}d{x}^{4}+8960\,{a}^{2}b{c}^{4}e{x}^{3}-17920\,{a}^{2}{c}^{5}d{x}^{3}+13440\,a{b}^{3}{c}^{3}e{x}^{3}-26880\,a{b}^{2}{c}^{4}d{x}^{3}+560\,{b}^{5}{c}^{2}e{x}^{3}-1120\,{b}^{4}{c}^{3}d{x}^{3}+13440\,{a}^{2}{b}^{2}{c}^{3}e{x}^{2}-26880\,{a}^{2}b{c}^{4}d{x}^{2}+2240\,a{b}^{4}{c}^{2}e{x}^{2}-4480\,a{b}^{3}{c}^{3}d{x}^{2}-56\,{b}^{6}ce{x}^{2}+112\,{b}^{5}{c}^{2}d{x}^{2}+4480\,{a}^{3}b{c}^{3}ex-8960\,{a}^{3}{c}^{4}dx+3360\,{a}^{2}{b}^{3}{c}^{2}ex-6720\,{a}^{2}{b}^{2}{c}^{3}dx-280\,a{b}^{5}cex+560\,a{b}^{4}{c}^{2}dx+14\,{b}^{7}ex-28\,{b}^{6}cdx+1280\,{a}^{4}{c}^{3}e+960\,{a}^{3}{b}^{2}{c}^{2}e-4480\,{a}^{3}b{c}^{3}d-80\,{a}^{2}{b}^{4}ce+1120\,{a}^{2}{b}^{3}{c}^{2}d+4\,a{b}^{6}e-168\,a{b}^{5}cd+10\,{b}^{7}d}{8960\,{a}^{4}{c}^{4}-8960\,{a}^{3}{b}^{2}{c}^{3}+3360\,{a}^{2}{b}^{4}{c}^{2}-560\,a{b}^{6}c+35\,{b}^{8}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(c*x^2+b*x+a)^(9/2),x)

[Out]

-2/35/(c*x^2+b*x+a)^(7/2)*(1024*b*c^6*e*x^7-2048*c^7*d*x^7+3584*b^2*c^5*e*x^6-7168*b*c^6*d*x^6+3584*a*b*c^5*e*
x^5-7168*a*c^6*d*x^5+4480*b^3*c^4*e*x^5-8960*b^2*c^5*d*x^5+8960*a*b^2*c^4*e*x^4-17920*a*b*c^5*d*x^4+2240*b^4*c
^3*e*x^4-4480*b^3*c^4*d*x^4+4480*a^2*b*c^4*e*x^3-8960*a^2*c^5*d*x^3+6720*a*b^3*c^3*e*x^3-13440*a*b^2*c^4*d*x^3
+280*b^5*c^2*e*x^3-560*b^4*c^3*d*x^3+6720*a^2*b^2*c^3*e*x^2-13440*a^2*b*c^4*d*x^2+1120*a*b^4*c^2*e*x^2-2240*a*
b^3*c^3*d*x^2-28*b^6*c*e*x^2+56*b^5*c^2*d*x^2+2240*a^3*b*c^3*e*x-4480*a^3*c^4*d*x+1680*a^2*b^3*c^2*e*x-3360*a^
2*b^2*c^3*d*x-140*a*b^5*c*e*x+280*a*b^4*c^2*d*x+7*b^7*e*x-14*b^6*c*d*x+640*a^4*c^3*e+480*a^3*b^2*c^2*e-2240*a^
3*b*c^3*d-40*a^2*b^4*c*e+560*a^2*b^3*c^2*d+2*a*b^6*e-84*a*b^5*c*d+5*b^7*d)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2
*b^4*c^2-16*a*b^6*c+b^8)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 168.243, size = 2043, normalized size = 11.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/35*(1024*(2*c^7*d - b*c^6*e)*x^7 + 3584*(2*b*c^6*d - b^2*c^5*e)*x^6 + 896*(2*(5*b^2*c^5 + 4*a*c^6)*d - (5*b^
3*c^4 + 4*a*b*c^5)*e)*x^5 + 2240*(2*(b^3*c^4 + 4*a*b*c^5)*d - (b^4*c^3 + 4*a*b^2*c^4)*e)*x^4 + 280*(2*(b^4*c^3
+ 24*a*b^2*c^4 + 16*a^2*c^5)*d - (b^5*c^2 + 24*a*b^3*c^3 + 16*a^2*b*c^4)*e)*x^3 - 28*(2*(b^5*c^2 - 40*a*b^3*c
^3 - 240*a^2*b*c^4)*d - (b^6*c - 40*a*b^4*c^2 - 240*a^2*b^2*c^3)*e)*x^2 - (5*b^7 - 84*a*b^5*c + 560*a^2*b^3*c^
2 - 2240*a^3*b*c^3)*d - 2*(a*b^6 - 20*a^2*b^4*c + 240*a^3*b^2*c^2 + 320*a^4*c^3)*e + 7*(2*(b^6*c - 20*a*b^4*c^
2 + 240*a^2*b^2*c^3 + 320*a^3*c^4)*d - (b^7 - 20*a*b^5*c + 240*a^2*b^3*c^2 + 320*a^3*b*c^3)*e)*x)*sqrt(c*x^2 +
b*x + a)/(a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3 + 256*a^8*c^4 + (b^8*c^4 - 16*a*b^6*c^5 +
96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^
3*c^6 + 256*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256*a^2*b^6*c^4 - 576*a^3*b^4*c^5 + 256*a^4*b^2*c^
6 + 512*a^5*c^7)*x^6 + 4*(b^11*c - 13*a*b^9*c^2 + 48*a^2*b^7*c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a^5*
b*c^6)*x^5 + (b^12 - 4*a*b^10*c - 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 153
6*a^6*c^6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*a^3*b^7*c^2 + 32*a^4*b^5*c^3 - 512*a^5*b^3*c^4 + 768*a^6*b*c^5)
*x^3 + 2*(3*a^2*b^10 - 46*a^3*b^8*c + 256*a^4*b^6*c^2 - 576*a^5*b^4*c^3 + 256*a^6*b^2*c^4 + 512*a^7*c^5)*x^2 +
4*(a^3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+b*x+a)**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.23998, size = 1129, normalized size = 6.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^(9/2),x, algorithm="giac")

[Out]

1/35*((4*(2*(8*(2*(4*(2*(2*c^7*d - b*c^6*e)*x/(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256
*a^4*c^8) + 7*(2*b*c^6*d - b^2*c^5*e)/(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8
))*x + 7*(10*b^2*c^5*d + 8*a*c^6*d - 5*b^3*c^4*e - 4*a*b*c^5*e)/(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256
*a^3*b^2*c^7 + 256*a^4*c^8))*x + 35*(2*b^3*c^4*d + 8*a*b*c^5*d - b^4*c^3*e - 4*a*b^2*c^4*e)/(b^8*c^4 - 16*a*b^
6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8))*x + 35*(2*b^4*c^3*d + 48*a*b^2*c^4*d + 32*a^2*c^5*d -
b^5*c^2*e - 24*a*b^3*c^3*e - 16*a^2*b*c^4*e)/(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256
*a^4*c^8))*x - 7*(2*b^5*c^2*d - 80*a*b^3*c^3*d - 480*a^2*b*c^4*d - b^6*c*e + 40*a*b^4*c^2*e + 240*a^2*b^2*c^3*
e)/(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8))*x + 7*(2*b^6*c*d - 40*a*b^4*c^2*
d + 480*a^2*b^2*c^3*d + 640*a^3*c^4*d - b^7*e + 20*a*b^5*c*e - 240*a^2*b^3*c^2*e - 320*a^3*b*c^3*e)/(b^8*c^4 -
16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8))*x - (5*b^7*d - 84*a*b^5*c*d + 560*a^2*b^3*c^2
*d - 2240*a^3*b*c^3*d + 2*a*b^6*e - 40*a^2*b^4*c*e + 480*a^3*b^2*c^2*e + 640*a^4*c^3*e)/(b^8*c^4 - 16*a*b^6*c^
5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8))/(c*x^2 + b*x + a)^(7/2)