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3.948 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=288 \[ \frac{5 \left (b^2-4 a c\right )^3 \left (-4 a A c-16 a b B+9 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{32768 a^{11/2}}-\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-16 a b B+9 A b^2\right )}{16384 a^5 x^2}+\frac{5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{6144 a^4 x^4}-\frac{(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{384 a^3 x^6}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8} \]

[Out]

(-5*(b^2 - 4*a*c)^2*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*
x^2])/(16384*a^5*x^2) + (5*(b^2 - 4*a*c)*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b
*x)*(a + b*x + c*x^2)^(3/2))/(6144*a^4*x^4) - ((9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2
*a + b*x)*(a + b*x + c*x^2)^(5/2))/(384*a^3*x^6) - (A*(a + b*x + c*x^2)^(7/2))/(
8*a*x^8) + ((9*A*b - 16*a*B)*(a + b*x + c*x^2)^(7/2))/(112*a^2*x^7) + (5*(b^2 -
4*a*c)^3*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a +
b*x + c*x^2])])/(32768*a^(11/2))

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Rubi [A]  time = 0.577456, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{5 \left (b^2-4 a c\right )^3 \left (-4 a A c-16 a b B+9 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{32768 a^{11/2}}-\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-16 a b B+9 A b^2\right )}{16384 a^5 x^2}+\frac{5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{6144 a^4 x^4}-\frac{(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a A c-16 a b B+9 A b^2\right )}{384 a^3 x^6}+\frac{(9 A b-16 a B) \left (a+b x+c x^2\right )^{7/2}}{112 a^2 x^7}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9,x]

[Out]

(-5*(b^2 - 4*a*c)^2*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*
x^2])/(16384*a^5*x^2) + (5*(b^2 - 4*a*c)*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2*a + b
*x)*(a + b*x + c*x^2)^(3/2))/(6144*a^4*x^4) - ((9*A*b^2 - 16*a*b*B - 4*a*A*c)*(2
*a + b*x)*(a + b*x + c*x^2)^(5/2))/(384*a^3*x^6) - (A*(a + b*x + c*x^2)^(7/2))/(
8*a*x^8) + ((9*A*b - 16*a*B)*(a + b*x + c*x^2)^(7/2))/(112*a^2*x^7) + (5*(b^2 -
4*a*c)^3*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a +
b*x + c*x^2])])/(32768*a^(11/2))

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Rubi in Sympy [A]  time = 63.9533, size = 287, normalized size = 1. \[ - \frac{A \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{8 a x^{8}} + \frac{\left (9 A b - 16 B a\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{112 a^{2} x^{7}} - \frac{\left (2 a + b x\right ) \left (- 4 A a c + b \left (9 A b - 16 B a\right )\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{384 a^{3} x^{6}} + \frac{5 \left (2 a + b x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 4 A a c + 9 A b^{2} - 16 B a b\right )}{6144 a^{4} x^{4}} - \frac{5 \left (2 a + b x\right ) \left (- 4 a c + b^{2}\right )^{2} \left (- 4 A a c + b \left (9 A b - 16 B a\right )\right ) \sqrt{a + b x + c x^{2}}}{16384 a^{5} x^{2}} + \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (- 4 A a c + 9 A b^{2} - 16 B a b\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{32768 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**9,x)

[Out]

-A*(a + b*x + c*x**2)**(7/2)/(8*a*x**8) + (9*A*b - 16*B*a)*(a + b*x + c*x**2)**(
7/2)/(112*a**2*x**7) - (2*a + b*x)*(-4*A*a*c + b*(9*A*b - 16*B*a))*(a + b*x + c*
x**2)**(5/2)/(384*a**3*x**6) + 5*(2*a + b*x)*(-4*a*c + b**2)*(a + b*x + c*x**2)*
*(3/2)*(-4*A*a*c + 9*A*b**2 - 16*B*a*b)/(6144*a**4*x**4) - 5*(2*a + b*x)*(-4*a*c
 + b**2)**2*(-4*A*a*c + b*(9*A*b - 16*B*a))*sqrt(a + b*x + c*x**2)/(16384*a**5*x
**2) + 5*(-4*a*c + b**2)**3*(-4*A*a*c + 9*A*b**2 - 16*B*a*b)*atanh((2*a + b*x)/(
2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(32768*a**(11/2))

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Mathematica [A]  time = 0.988211, size = 435, normalized size = 1.51 \[ \frac{-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (6144 a^7 (7 A+8 B x)+1024 a^6 x (A (99 b+119 c x)+4 B x (29 b+36 c x))+256 a^5 x^2 \left (A \left (243 b^2+614 b c x+413 c^2 x^2\right )+4 B x \left (74 b^2+197 b c x+144 c^2 x^2\right )\right )+384 a^4 x^3 \left (A \left (b^3+9 b^2 c x+29 b c^2 x^2+35 c^3 x^3\right )+2 B x \left (b^3+10 b^2 c x+38 b c^2 x^2+64 c^3 x^3\right )\right )-16 a^3 b x^4 \left (A \left (27 b^3+284 b^2 c x+1194 b c^2 x^2+2652 c^3 x^3\right )+56 b B x \left (b^2+12 b c x+66 c^2 x^2\right )\right )+56 a^2 b^3 x^5 \left (A \left (9 b^2+113 b c x+674 c^2 x^2\right )+20 b B x (b+16 c x)\right )-210 a b^5 x^6 (3 A b+50 A c x+8 b B x)+945 A b^7 x^7\right )-105 x^8 \log (x) \left (b^2-4 a c\right )^3 \left (-4 a A c-16 a b B+9 A b^2\right )+105 x^8 \left (b^2-4 a c\right )^3 \left (-4 a A c-16 a b B+9 A b^2\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{688128 a^{11/2} x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^9,x]

[Out]

(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(945*A*b^7*x^7 + 6144*a^7*(7*A + 8*B*x) - 210*
a*b^5*x^6*(3*A*b + 8*b*B*x + 50*A*c*x) + 1024*a^6*x*(4*B*x*(29*b + 36*c*x) + A*(
99*b + 119*c*x)) + 256*a^5*x^2*(4*B*x*(74*b^2 + 197*b*c*x + 144*c^2*x^2) + A*(24
3*b^2 + 614*b*c*x + 413*c^2*x^2)) + 56*a^2*b^3*x^5*(20*b*B*x*(b + 16*c*x) + A*(9
*b^2 + 113*b*c*x + 674*c^2*x^2)) + 384*a^4*x^3*(A*(b^3 + 9*b^2*c*x + 29*b*c^2*x^
2 + 35*c^3*x^3) + 2*B*x*(b^3 + 10*b^2*c*x + 38*b*c^2*x^2 + 64*c^3*x^3)) - 16*a^3
*b*x^4*(56*b*B*x*(b^2 + 12*b*c*x + 66*c^2*x^2) + A*(27*b^3 + 284*b^2*c*x + 1194*
b*c^2*x^2 + 2652*c^3*x^3))) - 105*(b^2 - 4*a*c)^3*(9*A*b^2 - 16*a*b*B - 4*a*A*c)
*x^8*Log[x] + 105*(b^2 - 4*a*c)^3*(9*A*b^2 - 16*a*b*B - 4*a*A*c)*x^8*Log[2*a + b
*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(688128*a^(11/2)*x^8)

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Maple [B]  time = 0.075, size = 2263, normalized size = 7.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^9,x)

[Out]

-235/6144*A*b^4/a^6*c^2*(c*x^2+b*x+a)^(5/2)+185/12288*A*b^5/a^6*c^2*(c*x^2+b*x+a
)^(3/2)*x-55/6144*A*b^4/a^6*c/x^2*(c*x^2+b*x+a)^(7/2)-15/128*B*b^3/a^(5/2)*c^2*l
n((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+15/512*B*b^5/a^(7/2)*c*ln((2*a+b*x+
2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+5/256*A/a^5*c^4*b*(c*x^2+b*x+a)^(5/2)*x+13/768
*A*b^3/a^5*c/x^3*(c*x^2+b*x+a)^(7/2)+1/12*B*b/a^2/x^6*(c*x^2+b*x+a)^(7/2)-45/163
84*A*b^7/a^6*(c*x^2+b*x+a)^(1/2)*x*c+5/64*B*b^2/a^3*c^3*(c*x^2+b*x+a)^(1/2)*x+5/
1024*B*b^6/a^5*(c*x^2+b*x+a)^(1/2)*x*c-1/128*A/a^4*c^2*b/x^3*(c*x^2+b*x+a)^(7/2)
+5/256*A/a^4*c^4*b*(c*x^2+b*x+a)^(3/2)*x-5/256*A/a^5*c^3*b/x*(c*x^2+b*x+a)^(7/2)
+5/256*A/a^3*c^4*b*(c*x^2+b*x+a)^(1/2)*x-1/8*A*(c*x^2+b*x+a)^(7/2)/a/x^8+1/192*A
/a^3*c^2/x^4*(c*x^2+b*x+a)^(7/2)+1/48*A/a^2*c/x^6*(c*x^2+b*x+a)^(7/2)+75/1024*A*
b^4/a^(7/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-35/2048*A*b^6/a^(9
/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15/128*A*b^2/a^(5/2)*c^3*ln(
(2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-305/6144*A*b^4/a^5*c^2*(c*x^2+b*x+a)^
(3/2)+1/32*B*b/a^4*c^2/x^2*(c*x^2+b*x+a)^(7/2)+1/48*B*b/a^3*c/x^4*(c*x^2+b*x+a)^
(7/2)-5/192*B*b^4/a^5*c^2*(c*x^2+b*x+a)^(3/2)*x+5/384*B*b^4/a^6*c/x*(c*x^2+b*x+a
)^(7/2)-5/384*B*b^4/a^6*c^2*(c*x^2+b*x+a)^(5/2)*x-5/128*B*b^4/a^4*c^2*(c*x^2+b*x
+a)^(1/2)*x-15/16384*A*b^7/a^7*c*(c*x^2+b*x+a)^(3/2)*x-9/16384*A*b^7/a^8*c*(c*x^
2+b*x+a)^(5/2)*x-7/512*A*b^2/a^5*c^2/x^2*(c*x^2+b*x+a)^(7/2)-31/4096*A*b^5/a^7*c
/x*(c*x^2+b*x+a)^(7/2)+31/4096*A*b^5/a^7*c^2*(c*x^2+b*x+a)^(5/2)*x+95/4096*A*b^5
/a^5*c^2*(c*x^2+b*x+a)^(1/2)*x+1/64*B*b^3/a^5*c/x^2*(c*x^2+b*x+a)^(7/2)-1/32*B*b
^2/a^4*c/x^3*(c*x^2+b*x+a)^(7/2)+5/64*B*b^2/a^4*c^3*(c*x^2+b*x+a)^(3/2)*x+5/3072
*B*b^6/a^6*c*(c*x^2+b*x+a)^(3/2)*x+1/1024*B*b^6/a^7*c*(c*x^2+b*x+a)^(5/2)*x+5/30
72*B*b^7/a^6*(c*x^2+b*x+a)^(3/2)+1/1024*B*b^7/a^7*(c*x^2+b*x+a)^(5/2)+5/1024*B*b
^7/a^5*(c*x^2+b*x+a)^(1/2)-1/7*B/a/x^7*(c*x^2+b*x+a)^(7/2)-5/2048*B*b^7/a^(9/2)*
ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-15/16384*A*b^8/a^7*(c*x^2+b*x+a)^(
3/2)-9/16384*A*b^8/a^8*(c*x^2+b*x+a)^(5/2)-45/16384*A*b^8/a^6*(c*x^2+b*x+a)^(1/2
)-1/128*A/a^4*c^4*(c*x^2+b*x+a)^(5/2)-5/384*A/a^3*c^4*(c*x^2+b*x+a)^(3/2)-5/128*
A/a^2*c^4*(c*x^2+b*x+a)^(1/2)+5/128*A/a^(3/2)*c^4*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b
*x+a)^(1/2))/x)+45/32768*A*b^8/a^(11/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2
))/x)-155/3072*A*b^3/a^5*c^3*(c*x^2+b*x+a)^(3/2)*x+145/3072*A*b^3/a^6*c^2/x*(c*x
^2+b*x+a)^(7/2)-145/3072*A*b^3/a^6*c^3*(c*x^2+b*x+a)^(5/2)*x-55/1024*A*b^3/a^4*c
^3*(c*x^2+b*x+a)^(1/2)*x-1/128*A*b^2/a^4*c/x^4*(c*x^2+b*x+a)^(7/2)-5/64*B*b^2/a^
5*c^2/x*(c*x^2+b*x+a)^(7/2)+5/64*B*b^2/a^5*c^3*(c*x^2+b*x+a)^(5/2)*x+5/32*B*b/a^
(3/2)*c^3*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/16*B*b^3/a^5*c^2*(c*x^
2+b*x+a)^(5/2)-1/24*B*b^2/a^3/x^5*(c*x^2+b*x+a)^(7/2)+5/64*B*b^3/a^4*c^2*(c*x^2+
b*x+a)^(3/2)+5/32*B*b^3/a^3*c^2*(c*x^2+b*x+a)^(1/2)+1/64*B*b^3/a^4/x^4*(c*x^2+b*
x+a)^(7/2)-1/384*B*b^4/a^5/x^3*(c*x^2+b*x+a)^(7/2)-25/512*B*b^5/a^4*c*(c*x^2+b*x
+a)^(1/2)-19/1536*B*b^5/a^6*c*(c*x^2+b*x+a)^(5/2)-35/1536*B*b^5/a^5*c*(c*x^2+b*x
+a)^(3/2)-1/1024*B*b^6/a^7/x*(c*x^2+b*x+a)^(7/2)-1/1536*B*b^5/a^6/x^2*(c*x^2+b*x
+a)^(7/2)-1/32*B*b/a^4*c^3*(c*x^2+b*x+a)^(5/2)-5/96*B*b/a^3*c^3*(c*x^2+b*x+a)^(3
/2)-5/32*B*b/a^2*c^3*(c*x^2+b*x+a)^(1/2)-205/2048*A*b^4/a^4*c^2*(c*x^2+b*x+a)^(1
/2)-9/1024*A*b^4/a^5/x^4*(c*x^2+b*x+a)^(7/2)+3/2048*A*b^5/a^6/x^3*(c*x^2+b*x+a)^
(7/2)+235/8192*A*b^6/a^5*c*(c*x^2+b*x+a)^(1/2)+59/8192*A*b^6/a^7*c*(c*x^2+b*x+a)
^(5/2)+325/24576*A*b^6/a^6*c*(c*x^2+b*x+a)^(3/2)+9/112*A*b/a^2/x^7*(c*x^2+b*x+a)
^(7/2)+9/16384*A*b^7/a^8/x*(c*x^2+b*x+a)^(7/2)+3/8192*A*b^6/a^7/x^2*(c*x^2+b*x+a
)^(7/2)+17/512*A*b^2/a^5*c^3*(c*x^2+b*x+a)^(5/2)+25/512*A*b^2/a^4*c^3*(c*x^2+b*x
+a)^(3/2)+65/512*A*b^2/a^3*c^3*(c*x^2+b*x+a)^(1/2)-3/64*A*b^2/a^3/x^6*(c*x^2+b*x
+a)^(7/2)+3/128*A*b^3/a^4/x^5*(c*x^2+b*x+a)^(7/2)+1/128*A/a^4*c^3/x^2*(c*x^2+b*x
+a)^(7/2)-1/96*A/a^3*c*b/x^5*(c*x^2+b*x+a)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.585353, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)/x^9,x, algorithm="fricas")

[Out]

[-1/1376256*(105*(16*B*a*b^7 - 9*A*b^8 - 256*A*a^4*c^4 - 256*(4*B*a^4*b - 3*A*a^
3*b^2)*c^3 + 96*(8*B*a^3*b^3 - 5*A*a^2*b^4)*c^2 - 16*(12*B*a^2*b^5 - 7*A*a*b^6)*
c)*x^8*log(-(4*(a*b*x + 2*a^2)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2 + 4*a*c)*
x^2 + 8*a^2)*sqrt(a))/x^2) + 4*(43008*A*a^7 - (1680*B*a*b^6 - 945*A*b^7 - 192*(2
56*B*a^4 - 221*A*a^3*b)*c^3 + 112*(528*B*a^3*b^2 - 337*A*a^2*b^3)*c^2 - 140*(128
*B*a^2*b^4 - 75*A*a*b^5)*c)*x^7 + 2*(560*B*a^2*b^5 - 315*A*a*b^6 + 6720*A*a^4*c^
3 + 48*(304*B*a^4*b - 199*A*a^3*b^2)*c^2 - 28*(192*B*a^3*b^3 - 113*A*a^2*b^4)*c)
*x^6 - 8*(112*B*a^3*b^4 - 63*A*a^2*b^5 - 48*(384*B*a^5 + 29*A*a^4*b)*c^2 - 8*(12
0*B*a^4*b^2 - 71*A*a^3*b^3)*c)*x^5 + 16*(48*B*a^4*b^3 - 27*A*a^3*b^4 + 6608*A*a^
5*c^2 + 8*(1576*B*a^5*b + 27*A*a^4*b^2)*c)*x^4 + 128*(592*B*a^5*b^2 + 3*A*a^4*b^
3 + 4*(288*B*a^6 + 307*A*a^5*b)*c)*x^3 + 256*(464*B*a^6*b + 243*A*a^5*b^2 + 476*
A*a^6*c)*x^2 + 3072*(16*B*a^7 + 33*A*a^6*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a
^(11/2)*x^8), -1/688128*(105*(16*B*a*b^7 - 9*A*b^8 - 256*A*a^4*c^4 - 256*(4*B*a^
4*b - 3*A*a^3*b^2)*c^3 + 96*(8*B*a^3*b^3 - 5*A*a^2*b^4)*c^2 - 16*(12*B*a^2*b^5 -
 7*A*a*b^6)*c)*x^8*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) +
2*(43008*A*a^7 - (1680*B*a*b^6 - 945*A*b^7 - 192*(256*B*a^4 - 221*A*a^3*b)*c^3 +
 112*(528*B*a^3*b^2 - 337*A*a^2*b^3)*c^2 - 140*(128*B*a^2*b^4 - 75*A*a*b^5)*c)*x
^7 + 2*(560*B*a^2*b^5 - 315*A*a*b^6 + 6720*A*a^4*c^3 + 48*(304*B*a^4*b - 199*A*a
^3*b^2)*c^2 - 28*(192*B*a^3*b^3 - 113*A*a^2*b^4)*c)*x^6 - 8*(112*B*a^3*b^4 - 63*
A*a^2*b^5 - 48*(384*B*a^5 + 29*A*a^4*b)*c^2 - 8*(120*B*a^4*b^2 - 71*A*a^3*b^3)*c
)*x^5 + 16*(48*B*a^4*b^3 - 27*A*a^3*b^4 + 6608*A*a^5*c^2 + 8*(1576*B*a^5*b + 27*
A*a^4*b^2)*c)*x^4 + 128*(592*B*a^5*b^2 + 3*A*a^4*b^3 + 4*(288*B*a^6 + 307*A*a^5*
b)*c)*x^3 + 256*(464*B*a^6*b + 243*A*a^5*b^2 + 476*A*a^6*c)*x^2 + 3072*(16*B*a^7
 + 33*A*a^6*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^5*x^8)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**9,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.327431, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)/x^9,x, algorithm="giac")

[Out]

Done

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