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]

-2/7*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(7/2)+24/35*(-b*e+2*c*d)*(2*c*x+b)/(-4*a*c+b^2)^2/(
c*x^2+b*x+a)^(5/2)-128/35*c*(-b*e+2*c*d)*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^(3/2)+1024/35*c^2*(-b*e+2*c*d)
*(2*c*x+b)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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 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 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]

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*}

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Mathematica [A]  time = 0.19, 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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fricas [B]  time = 41.49, size = 941, normalized size = 5.20 \[ \frac {2 \, {\left (1024 \, {\left (2 \, c^{7} d - b c^{6} e\right )} x^{7} + 3584 \, {\left (2 \, b c^{6} d - b^{2} c^{5} e\right )} x^{6} + 896 \, {\left (2 \, {\left (5 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d - {\left (5 \, b^{3} c^{4} + 4 \, a b c^{5}\right )} e\right )} x^{5} + 2240 \, {\left (2 \, {\left (b^{3} c^{4} + 4 \, a b c^{5}\right )} d - {\left (b^{4} c^{3} + 4 \, a b^{2} c^{4}\right )} e\right )} x^{4} + 280 \, {\left (2 \, {\left (b^{4} c^{3} + 24 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d - {\left (b^{5} c^{2} + 24 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} e\right )} x^{3} - 28 \, {\left (2 \, {\left (b^{5} c^{2} - 40 \, a b^{3} c^{3} - 240 \, a^{2} b c^{4}\right )} d - {\left (b^{6} c - 40 \, a b^{4} c^{2} - 240 \, a^{2} b^{2} c^{3}\right )} e\right )} x^{2} - {\left (5 \, b^{7} - 84 \, a b^{5} c + 560 \, a^{2} b^{3} c^{2} - 2240 \, a^{3} b c^{3}\right )} d - 2 \, {\left (a b^{6} - 20 \, a^{2} b^{4} c + 240 \, a^{3} b^{2} c^{2} + 320 \, a^{4} c^{3}\right )} e + 7 \, {\left (2 \, {\left (b^{6} c - 20 \, a b^{4} c^{2} + 240 \, a^{2} b^{2} c^{3} + 320 \, a^{3} c^{4}\right )} d - {\left (b^{7} - 20 \, a b^{5} c + 240 \, a^{2} b^{3} c^{2} + 320 \, a^{3} b c^{3}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{35 \, {\left (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} + {\left (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}\right )} x^{8} + 4 \, {\left (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}\right )} x^{7} + 2 \, {\left (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}\right )} x^{6} + 4 \, {\left (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}\right )} x^{5} + {\left (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} + 1536 \, a^{6} c^{6}\right )} x^{4} + 4 \, {\left (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}\right )} x^{3} + 2 \, {\left (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}\right )} x^{2} + 4 \, {\left (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}\right )} x\right )}} \]

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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giac [B]  time = 0.32, size = 788, normalized size = 4.35 \[ \frac {2 \, {\left ({\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (2 \, c^{7} d - b c^{6} e\right )} x}{b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}} + \frac {7 \, {\left (2 \, b c^{6} d - b^{2} c^{5} e\right )}}{b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}}\right )} x + \frac {7 \, {\left (10 \, b^{2} c^{5} d + 8 \, a c^{6} d - 5 \, b^{3} c^{4} e - 4 \, a b c^{5} e\right )}}{b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}}\right )} x + \frac {35 \, {\left (2 \, b^{3} c^{4} d + 8 \, a b c^{5} d - b^{4} c^{3} e - 4 \, a b^{2} c^{4} e\right )}}{b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}}\right )} x + \frac {35 \, {\left (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\right )}}{b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}}\right )} x - \frac {7 \, {\left (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\right )}}{b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}}\right )} x + \frac {7 \, {\left (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\right )}}{b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}}\right )} x - \frac {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} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}}\right )}}{35 \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}}} \]

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]

2/35*((4*(2*(8*(2*(4*(2*(2*c^7*d - b*c^6*e)*x/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c
^4) + 7*(2*b*c^6*d - b^2*c^5*e)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*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 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256
*a^4*c^4))*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 - 16*a*b^6*c + 96*a^2*b^4*c^2 -
256*a^3*b^2*c^3 + 256*a^4*c^4))*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 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*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 - 16*a*b^6*c + 96*a^2*
b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*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 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b
^2*c^3 + 256*a^4*c^4))*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 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c
^4))/(c*x^2 + b*x + a)^(7/2)

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maple [B]  time = 0.06, size = 500, normalized size = 2.76 \[ -\frac {2 \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+3584 a b \,c^{5} e \,x^{5}-7168 a \,c^{6} d \,x^{5}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+8960 a \,b^{2} c^{4} e \,x^{4}-17920 a b \,c^{5} d \,x^{4}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+4480 a^{2} b \,c^{4} e \,x^{3}-8960 a^{2} c^{5} d \,x^{3}+6720 a \,b^{3} c^{3} e \,x^{3}-13440 a \,b^{2} c^{4} d \,x^{3}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}+6720 a^{2} b^{2} c^{3} e \,x^{2}-13440 a^{2} b \,c^{4} d \,x^{2}+1120 a \,b^{4} c^{2} e \,x^{2}-2240 a \,b^{3} c^{3} d \,x^{2}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+2240 a^{3} b \,c^{3} e x -4480 a^{3} c^{4} d x +1680 a^{2} b^{3} c^{2} e x -3360 a^{2} b^{2} c^{3} d x -140 a \,b^{5} c e x +280 a \,b^{4} c^{2} d x +7 b^{7} e x -14 b^{6} c d x +640 a^{4} c^{3} e +480 a^{3} b^{2} c^{2} e -2240 a^{3} b \,c^{3} d -40 a^{2} b^{4} c e +560 a^{2} b^{3} c^{2} d +2 a \,b^{6} e -84 a \,b^{5} c d +5 b^{7} d \right )}{35 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 2.47, size = 599, normalized size = 3.31 \[ \frac {x\,\left (\frac {2\,c^2\,\left (768\,c^2\,d-368\,b\,c\,e\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}-\frac {32\,b\,c^3\,e}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}\right )+\frac {b\,c\,\left (768\,c^2\,d-368\,b\,c\,e\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}-\frac {64\,a\,c^3\,e}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}+\frac {x\,\left (\frac {4\,c^2\,d}{7\,\left (4\,a\,c^2-b^2\,c\right )}-\frac {2\,b\,c\,e}{7\,\left (4\,a\,c^2-b^2\,c\right )}\right )-\frac {4\,a\,c\,e}{7\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,b\,c\,d}{7\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{7/2}}-\frac {x\,\left (\frac {2\,c^2\,\left (28\,b\,e-48\,c\,d\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,b\,c^2\,e}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {b\,c\,\left (28\,b\,e-48\,c\,d\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {16\,a\,c^2\,e}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {\frac {2\,c^2\,x\,\left (2048\,c^3\,d-1024\,b\,c^2\,e\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^3}+\frac {b\,c\,\left (2048\,c^3\,d-1024\,b\,c^2\,e\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^3}}{\sqrt {c\,x^2+b\,x+a}}-\frac {4\,e}{\left (140\,a\,c-35\,b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {16\,c\,e}{105\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*((2*c^2*(768*c^2*d - 368*b*c*e))/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) - (32*b*c^3*e)/(105*(4*a*c^2 - b^2
*c)*(4*a*c - b^2)^2)) + (b*c*(768*c^2*d - 368*b*c*e))/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) - (64*a*c^3*e)/(
105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2))/(a + b*x + c*x^2)^(3/2) + (x*((4*c^2*d)/(7*(4*a*c^2 - b^2*c)) - (2*b*c
*e)/(7*(4*a*c^2 - b^2*c))) - (4*a*c*e)/(7*(4*a*c^2 - b^2*c)) + (2*b*c*d)/(7*(4*a*c^2 - b^2*c)))/(a + b*x + c*x
^2)^(7/2) - (x*((2*c^2*(28*b*e - 48*c*d))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c^2*e)/(35*(4*a*c^2 - b^
2*c)*(4*a*c - b^2))) + (b*c*(28*b*e - 48*c*d))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (16*a*c^2*e)/(35*(4*a*c^
2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(5/2) + ((2*c^2*x*(2048*c^3*d - 1024*b*c^2*e))/(35*(4*a*c^2 - b^2
*c)*(4*a*c - b^2)^3) + (b*c*(2048*c^3*d - 1024*b*c^2*e))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^3))/(a + b*x + c*
x^2)^(1/2) - (4*e)/((140*a*c - 35*b^2)*(a + b*x + c*x^2)^(5/2)) + (16*c*e)/(105*(4*a*c - b^2)^2*(a + b*x + c*x
^2)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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