3.949 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=375 \[ \frac{5 \left (b^2-4 a c\right )^3 \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{65536 a^{13/2}}-\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt{a+b x+c x^2} \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )}{32768 a^6 x^2}+\frac{5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )}{12288 a^5 x^4}-\frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{2016 a^3 x^7}+\frac{(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}+\frac{(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right )}{768 a^4 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9} \]

[Out]

(-5*(b^2 - 4*a*c)^2*(2*a*B*(9*b^2 - 4*a*c) - A*(11*b^3 - 12*a*b*c))*(2*a + b*x)*
Sqrt[a + b*x + c*x^2])/(32768*a^6*x^2) + (5*(b^2 - 4*a*c)*(2*a*B*(9*b^2 - 4*a*c)
- A*(11*b^3 - 12*a*b*c))*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(12288*a^5*x^4) +
((11*A*b^3 - 18*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*(2*a + b*x)*(a + b*x + c*x^2)
^(5/2))/(768*a^4*x^6) - (A*(a + b*x + c*x^2)^(7/2))/(9*a*x^9) + ((11*A*b - 18*a*
B)*(a + b*x + c*x^2)^(7/2))/(144*a^2*x^8) - ((99*A*b^2 - 162*a*b*B - 64*a*A*c)*(
a + b*x + c*x^2)^(7/2))/(2016*a^3*x^7) + (5*(b^2 - 4*a*c)^3*(2*a*B*(9*b^2 - 4*a*
c) - A*(11*b^3 - 12*a*b*c))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2]
)])/(65536*a^(13/2))

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Rubi [A]  time = 1.06019, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{5 \left (b^2-4 a c\right )^2 (2 a+b x) \sqrt{a+b x+c x^2} \left (2 a B \left (9 b^2-4 a c\right )-A \left (11 b^3-12 a b c\right )\right )}{32768 a^6 x^2}-\frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a A c-162 a b B+99 A b^2\right )}{2016 a^3 x^7}+\frac{(11 A b-18 a B) \left (a+b x+c x^2\right )^{7/2}}{144 a^2 x^8}-\frac{5 \left (b^2-4 a c\right )^3 \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{65536 a^{13/2}}-\frac{5 \left (b^2-4 a c\right ) (2 a+b x) \left (a+b x+c x^2\right )^{3/2} \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right )}{12288 a^5 x^4}+\frac{(2 a+b x) \left (a+b x+c x^2\right )^{5/2} \left (8 a^2 B c-12 a A b c-18 a b^2 B+11 A b^3\right )}{768 a^4 x^6}-\frac{A \left (a+b x+c x^2\right )^{7/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^10,x]

[Out]

(-5*(b^2 - 4*a*c)^2*(2*a*B*(9*b^2 - 4*a*c) - A*(11*b^3 - 12*a*b*c))*(2*a + b*x)*
Sqrt[a + b*x + c*x^2])/(32768*a^6*x^2) - (5*(b^2 - 4*a*c)*(11*A*b^3 - 18*a*b^2*B
- 12*a*A*b*c + 8*a^2*B*c)*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(12288*a^5*x^4)
+ ((11*A*b^3 - 18*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*(2*a + b*x)*(a + b*x + c*x^2
)^(5/2))/(768*a^4*x^6) - (A*(a + b*x + c*x^2)^(7/2))/(9*a*x^9) + ((11*A*b - 18*a
*B)*(a + b*x + c*x^2)^(7/2))/(144*a^2*x^8) - ((99*A*b^2 - 162*a*b*B - 64*a*A*c)*
(a + b*x + c*x^2)^(7/2))/(2016*a^3*x^7) - (5*(b^2 - 4*a*c)^3*(11*A*b^3 - 18*a*b^
2*B - 12*a*A*b*c + 8*a^2*B*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^
2])])/(65536*a^(13/2))

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Rubi in Sympy [A]  time = 117.99, size = 381, normalized size = 1.02 \[ - \frac{A \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{9 a x^{9}} + \frac{\left (11 A b - 18 B a\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{144 a^{2} x^{8}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{7}{2}} \left (- 64 A a c + 99 A b^{2} - 162 B a b\right )}{2016 a^{3} x^{7}} + \frac{\left (2 a + b x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (- 12 A a b c + 11 A b^{3} + 8 B a^{2} c - 18 B a b^{2}\right )}{768 a^{4} x^{6}} - \frac{5 \left (2 a + b x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 12 A a b c + 11 A b^{3} + 8 B a^{2} c - 18 B a b^{2}\right )}{12288 a^{5} x^{4}} + \frac{5 \left (2 a + b x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (- 12 A a b c + 11 A b^{3} + 8 B a^{2} c - 18 B a b^{2}\right )}{32768 a^{6} x^{2}} - \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (- 12 A a b c + 11 A b^{3} + 8 B a^{2} c - 18 B a b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{65536 a^{\frac{13}{2}}} \]

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**10,x)

[Out]

-A*(a + b*x + c*x**2)**(7/2)/(9*a*x**9) + (11*A*b - 18*B*a)*(a + b*x + c*x**2)**
(7/2)/(144*a**2*x**8) - (a + b*x + c*x**2)**(7/2)*(-64*A*a*c + 99*A*b**2 - 162*B
*a*b)/(2016*a**3*x**7) + (2*a + b*x)*(a + b*x + c*x**2)**(5/2)*(-12*A*a*b*c + 11
*A*b**3 + 8*B*a**2*c - 18*B*a*b**2)/(768*a**4*x**6) - 5*(2*a + b*x)*(-4*a*c + b*
*2)*(a + b*x + c*x**2)**(3/2)*(-12*A*a*b*c + 11*A*b**3 + 8*B*a**2*c - 18*B*a*b**
2)/(12288*a**5*x**4) + 5*(2*a + b*x)*(-4*a*c + b**2)**2*sqrt(a + b*x + c*x**2)*(
-12*A*a*b*c + 11*A*b**3 + 8*B*a**2*c - 18*B*a*b**2)/(32768*a**6*x**2) - 5*(-4*a*
c + b**2)**3*(-12*A*a*b*c + 11*A*b**3 + 8*B*a**2*c - 18*B*a*b**2)*atanh((2*a + b
*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(65536*a**(13/2))

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Mathematica [A]  time = 1.18785, size = 554, normalized size = 1.48 \[ \frac{-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (28672 a^8 (8 A+9 B x)+2048 a^7 x (A (259 b+304 c x)+3 B x (99 b+119 c x))+1536 a^6 x^2 \left (A \left (206 b^2+502 b c x+320 c^2 x^2\right )+B x \left (243 b^2+614 b c x+413 c^2 x^2\right )\right )+256 a^5 x^3 \left (A \left (5 b^3+42 b^2 c x+123 b c^2 x^2+128 c^3 x^3\right )+9 B x \left (b^3+9 b^2 c x+29 b c^2 x^2+35 c^3 x^3\right )\right )-32 a^4 x^4 \left (4 A \left (11 b^4+107 b^3 c x+399 b^2 c^2 x^2+689 b c^3 x^3+512 c^4 x^4\right )+3 b B x \left (27 b^3+284 b^2 c x+1194 b c^2 x^2+2652 c^3 x^3\right )\right )+48 a^3 b^2 x^5 \left (3 A \left (11 b^3+124 b^2 c x+586 b c^2 x^2+1628 c^3 x^3\right )+7 b B x \left (9 b^2+113 b c x+674 c^2 x^2\right )\right )-84 a^2 b^4 x^6 \left (2 A \left (11 b^2+147 b c x+966 c^2 x^2\right )+15 b B x (3 b+50 c x)\right )+210 a b^6 x^7 (11 A b+194 A c x+27 b B x)-3465 A b^8 x^8\right )+315 x^9 \log (x) \left (b^2-4 a c\right )^3 \left (A \left (11 b^3-12 a b c\right )+2 a B \left (4 a c-9 b^2\right )\right )-315 x^9 \left (b^2-4 a c\right )^3 \left (A \left (11 b^3-12 a b c\right )+2 a B \left (4 a c-9 b^2\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{4128768 a^{13/2} x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^(5/2))/x^10,x]

[Out]

(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-3465*A*b^8*x^8 + 28672*a^8*(8*A + 9*B*x) + 2
10*a*b^6*x^7*(11*A*b + 27*b*B*x + 194*A*c*x) + 2048*a^7*x*(3*B*x*(99*b + 119*c*x
) + A*(259*b + 304*c*x)) + 1536*a^6*x^2*(A*(206*b^2 + 502*b*c*x + 320*c^2*x^2) +
B*x*(243*b^2 + 614*b*c*x + 413*c^2*x^2)) - 84*a^2*b^4*x^6*(15*b*B*x*(3*b + 50*c
*x) + 2*A*(11*b^2 + 147*b*c*x + 966*c^2*x^2)) + 256*a^5*x^3*(9*B*x*(b^3 + 9*b^2*
c*x + 29*b*c^2*x^2 + 35*c^3*x^3) + A*(5*b^3 + 42*b^2*c*x + 123*b*c^2*x^2 + 128*c
^3*x^3)) + 48*a^3*b^2*x^5*(7*b*B*x*(9*b^2 + 113*b*c*x + 674*c^2*x^2) + 3*A*(11*b
^3 + 124*b^2*c*x + 586*b*c^2*x^2 + 1628*c^3*x^3)) - 32*a^4*x^4*(3*b*B*x*(27*b^3
+ 284*b^2*c*x + 1194*b*c^2*x^2 + 2652*c^3*x^3) + 4*A*(11*b^4 + 107*b^3*c*x + 399
*b^2*c^2*x^2 + 689*b*c^3*x^3 + 512*c^4*x^4))) + 315*(b^2 - 4*a*c)^3*(2*a*B*(-9*b
^2 + 4*a*c) + A*(11*b^3 - 12*a*b*c))*x^9*Log[x] - 315*(b^2 - 4*a*c)^3*(2*a*B*(-9
*b^2 + 4*a*c) + A*(11*b^3 - 12*a*b*c))*x^9*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*
(b + c*x)]])/(4128768*a^(13/2)*x^9)

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Maple [B]  time = 0.108, size = 2677, normalized size = 7.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(5/2)/x^10,x)

[Out]

-55/1024*B*b^3/a^4*c^3*(c*x^2+b*x+a)^(1/2)*x+13/768*B*b^3/a^5*c/x^3*(c*x^2+b*x+a
)^(7/2)-155/3072*B*b^3/a^5*c^3*(c*x^2+b*x+a)^(3/2)*x-145/3072*B*b^3/a^6*c^3*(c*x
^2+b*x+a)^(5/2)*x+145/3072*B*b^3/a^6*c^2/x*(c*x^2+b*x+a)^(7/2)-1/96*B/a^3*c*b/x^
5*(c*x^2+b*x+a)^(7/2)+5/256*B/a^3*c^4*b*(c*x^2+b*x+a)^(1/2)*x-45/16384*B*b^7/a^6
*(c*x^2+b*x+a)^(1/2)*x*c-55/6144*B*b^4/a^6*c/x^2*(c*x^2+b*x+a)^(7/2)-15/16384*B*
b^7/a^7*c*(c*x^2+b*x+a)^(3/2)*x-15/512*A*b^2/a^5*c^4*(c*x^2+b*x+a)^(3/2)*x-1/128
*B/a^4*c^2*b/x^3*(c*x^2+b*x+a)^(7/2)+5/256*B/a^4*c^4*b*(c*x^2+b*x+a)^(3/2)*x+5/1
28*B/a^(3/2)*c^4*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-1/8*B/a/x^8*(c*x^
2+b*x+a)^(7/2)+45/32768*B*b^8/a^(11/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2)
)/x)+11/32768*A*b^9/a^9*(c*x^2+b*x+a)^(5/2)+55/32768*A*b^9/a^7*(c*x^2+b*x+a)^(1/
2)+55/98304*A*b^9/a^8*(c*x^2+b*x+a)^(3/2)-55/65536*A*b^9/a^(13/2)*ln((2*a+b*x+2*
a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-9/16384*B*b^8/a^8*(c*x^2+b*x+a)^(5/2)-45/16384*B
*b^8/a^6*(c*x^2+b*x+a)^(1/2)-15/16384*B*b^8/a^7*(c*x^2+b*x+a)^(3/2)-1/128*B/a^4*
c^4*(c*x^2+b*x+a)^(5/2)-5/128*B/a^2*c^4*(c*x^2+b*x+a)^(1/2)-5/384*B/a^3*c^4*(c*x
^2+b*x+a)^(3/2)-1/9*A*(c*x^2+b*x+a)^(7/2)/a/x^9-1/128*A*b/a^4*c^2/x^4*(c*x^2+b*x
+a)^(7/2)-3/256*A*b/a^5*c^3/x^2*(c*x^2+b*x+a)^(7/2)-1/32*A*b/a^3*c/x^6*(c*x^2+b*
x+a)^(7/2)-15/512*A*b^2/a^6*c^4*(c*x^2+b*x+a)^(5/2)*x+15/512*A*b^2/a^6*c^3/x*(c*
x^2+b*x+a)^(7/2)-115/1024*A*b^3/a^4*c^3*(c*x^2+b*x+a)^(1/2)-35/1024*A*b^3/a^6*c^
3*(c*x^2+b*x+a)^(5/2)+1/768*A*b^3/a^5*c/x^4*(c*x^2+b*x+a)^(7/2)-5/256*B/a^5*c^3*
b/x*(c*x^2+b*x+a)^(7/2)-1/128*B*b^2/a^4*c/x^4*(c*x^2+b*x+a)^(7/2)-7/512*B*b^2/a^
5*c^2/x^2*(c*x^2+b*x+a)^(7/2)+95/4096*B*b^5/a^5*c^2*(c*x^2+b*x+a)^(1/2)*x+185/12
288*B*b^5/a^6*c^2*(c*x^2+b*x+a)^(3/2)*x+31/4096*B*b^5/a^7*c^2*(c*x^2+b*x+a)^(5/2
)*x-31/4096*B*b^5/a^7*c/x*(c*x^2+b*x+a)^(7/2)+5/1024*A*b^3/a^6*c^2/x^2*(c*x^2+b*
x+a)^(7/2)-125/8192*A*b^6/a^6*c^2*(c*x^2+b*x+a)^(1/2)*x-235/24576*A*b^6/a^7*c^2*
(c*x^2+b*x+a)^(3/2)*x-119/24576*A*b^6/a^8*c^2*(c*x^2+b*x+a)^(5/2)*x+119/24576*A*
b^6/a^8*c/x*(c*x^2+b*x+a)^(7/2)+55/98304*A*b^8/a^8*c*(c*x^2+b*x+a)^(3/2)*x+11/32
768*A*b^8/a^9*c*(c*x^2+b*x+a)^(5/2)*x+55/32768*A*b^8/a^7*(c*x^2+b*x+a)^(1/2)*x*c
+23/4096*A*b^5/a^7*c/x^2*(c*x^2+b*x+a)^(7/2)+85/2048*A*b^4/a^5*c^3*(c*x^2+b*x+a)
^(1/2)*x-5/512*A*b^4/a^6*c/x^3*(c*x^2+b*x+a)^(7/2)+75/2048*A*b^4/a^6*c^3*(c*x^2+
b*x+a)^(3/2)*x+65/2048*A*b^4/a^7*c^3*(c*x^2+b*x+a)^(5/2)*x-65/2048*A*b^4/a^7*c^2
/x*(c*x^2+b*x+a)^(7/2)+1/64*A*b^2/a^4*c/x^5*(c*x^2+b*x+a)^(7/2)+3/256*A*b^2/a^5*
c^2/x^3*(c*x^2+b*x+a)^(7/2)-15/512*A*b^2/a^4*c^4*(c*x^2+b*x+a)^(1/2)*x+5/256*B/a
^5*c^4*b*(c*x^2+b*x+a)^(5/2)*x+1/128*B/a^4*c^3/x^2*(c*x^2+b*x+a)^(7/2)+1/48*B/a^
2*c/x^6*(c*x^2+b*x+a)^(7/2)+75/1024*B*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*B*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*B*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)
+9/112*B*b/a^2/x^7*(c*x^2+b*x+a)^(7/2)+3/128*B*b^3/a^4/x^5*(c*x^2+b*x+a)^(7/2)-2
35/6144*B*b^4/a^6*c^2*(c*x^2+b*x+a)^(5/2)-205/2048*B*b^4/a^4*c^2*(c*x^2+b*x+a)^(
1/2)-305/6144*B*b^4/a^5*c^2*(c*x^2+b*x+a)^(3/2)-9/1024*B*b^4/a^5/x^4*(c*x^2+b*x+
a)^(7/2)+59/8192*B*b^6/a^7*c*(c*x^2+b*x+a)^(5/2)+325/24576*B*b^6/a^6*c*(c*x^2+b*
x+a)^(3/2)+9/16384*B*b^7/a^8/x*(c*x^2+b*x+a)^(7/2)+3/8192*B*b^6/a^7/x^2*(c*x^2+b
*x+a)^(7/2)+235/8192*B*b^6/a^5*c*(c*x^2+b*x+a)^(1/2)+3/2048*B*b^5/a^6/x^3*(c*x^2
+b*x+a)^(7/2)+17/512*B*b^2/a^5*c^3*(c*x^2+b*x+a)^(5/2)+65/512*B*b^2/a^3*c^3*(c*x
^2+b*x+a)^(1/2)+25/512*B*b^2/a^4*c^3*(c*x^2+b*x+a)^(3/2)-145/3072*A*b^3/a^5*c^3*
(c*x^2+b*x+a)^(3/2)+11/384*A*b^3/a^4/x^6*(c*x^2+b*x+a)^(7/2)+3/256*A*b/a^5*c^4*(
c*x^2+b*x+a)^(5/2)+2/63*A/a^2*c/x^7*(c*x^2+b*x+a)^(7/2)-15/256*A*b/a^(5/2)*c^4*l
n((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-105/2048*A*b^5/a^(9/2)*c^2*ln((2*a+
b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+45/4096*A*b^7/a^(11/2)*c*ln((2*a+b*x+2*a^(
1/2)*(c*x^2+b*x+a)^(1/2))/x)+25/256*A*b^3/a^(7/2)*c^3*ln((2*a+b*x+2*a^(1/2)*(c*x
^2+b*x+a)^(1/2))/x)+15/256*A*b/a^3*c^4*(c*x^2+b*x+a)^(1/2)+5/256*A*b/a^4*c^4*(c*
x^2+b*x+a)^(3/2)+11/144*A*b/a^2/x^8*(c*x^2+b*x+a)^(7/2)-11/224*A*b^2/a^3/x^7*(c*
x^2+b*x+a)^(7/2)-11/768*A*b^4/a^5/x^5*(c*x^2+b*x+a)^(7/2)+107/4096*A*b^5/a^7*c^2
*(c*x^2+b*x+a)^(5/2)+295/4096*A*b^5/a^5*c^2*(c*x^2+b*x+a)^(1/2)+145/4096*A*b^5/a
^6*c^2*(c*x^2+b*x+a)^(3/2)+11/2048*A*b^5/a^6/x^4*(c*x^2+b*x+a)^(7/2)-227/49152*A
*b^7/a^8*c*(c*x^2+b*x+a)^(5/2)-415/49152*A*b^7/a^7*c*(c*x^2+b*x+a)^(3/2)-11/3276
8*A*b^8/a^9/x*(c*x^2+b*x+a)^(7/2)-11/49152*A*b^7/a^8/x^2*(c*x^2+b*x+a)^(7/2)-305
/16384*A*b^7/a^6*c*(c*x^2+b*x+a)^(1/2)-11/12288*A*b^6/a^7/x^3*(c*x^2+b*x+a)^(7/2
)-3/64*B*b^2/a^3/x^6*(c*x^2+b*x+a)^(7/2)+1/192*B/a^3*c^2/x^4*(c*x^2+b*x+a)^(7/2)
-9/16384*B*b^7/a^8*c*(c*x^2+b*x+a)^(5/2)*x

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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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.678367, 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^10,x, algorithm="fricas")

[Out]

[1/8257536*(315*(18*B*a*b^8 - 11*A*b^9 + 256*(2*B*a^5 - 3*A*a^4*b)*c^4 - 256*(6*
B*a^4*b^2 - 5*A*a^3*b^3)*c^3 + 96*(10*B*a^3*b^4 - 7*A*a^2*b^5)*c^2 - 16*(14*B*a^
2*b^6 - 9*A*a*b^7)*c)*x^9*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*(229376*A*a^8 + (5670*B*a*b^7
- 3465*A*b^8 - 65536*A*a^4*c^4 - 576*(442*B*a^4*b - 407*A*a^3*b^2)*c^3 + 336*(67
4*B*a^3*b^3 - 483*A*a^2*b^4)*c^2 - 420*(150*B*a^2*b^5 - 97*A*a*b^6)*c)*x^8 - 2*(
1890*B*a^2*b^6 - 1155*A*a*b^7 - 64*(630*B*a^5 - 689*A*a^4*b)*c^3 + 144*(398*B*a^
4*b^2 - 293*A*a^3*b^3)*c^2 - 84*(226*B*a^3*b^4 - 147*A*a^2*b^5)*c)*x^7 + 8*(378*
B*a^3*b^5 - 231*A*a^2*b^6 + 4096*A*a^5*c^3 + 48*(174*B*a^5*b - 133*A*a^4*b^2)*c^
2 - 24*(142*B*a^4*b^3 - 93*A*a^3*b^4)*c)*x^6 - 16*(162*B*a^4*b^4 - 99*A*a^3*b^5
- 48*(826*B*a^6 + 41*A*a^5*b)*c^2 - 8*(162*B*a^5*b^2 - 107*A*a^4*b^3)*c)*x^5 + 1
28*(18*B*a^5*b^3 - 11*A*a^4*b^4 + 3840*A*a^6*c^2 + 12*(614*B*a^6*b + 7*A*a^5*b^2
)*c)*x^4 + 256*(1458*B*a^6*b^2 + 5*A*a^5*b^3 + 12*(238*B*a^7 + 251*A*a^6*b)*c)*x
^3 + 1024*(594*B*a^7*b + 309*A*a^6*b^2 + 608*A*a^7*c)*x^2 + 14336*(18*B*a^8 + 37
*A*a^7*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(13/2)*x^9), 1/4128768*(315*(18*B
*a*b^8 - 11*A*b^9 + 256*(2*B*a^5 - 3*A*a^4*b)*c^4 - 256*(6*B*a^4*b^2 - 5*A*a^3*b
^3)*c^3 + 96*(10*B*a^3*b^4 - 7*A*a^2*b^5)*c^2 - 16*(14*B*a^2*b^6 - 9*A*a*b^7)*c)
*x^9*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) - 2*(229376*A*a^
8 + (5670*B*a*b^7 - 3465*A*b^8 - 65536*A*a^4*c^4 - 576*(442*B*a^4*b - 407*A*a^3*
b^2)*c^3 + 336*(674*B*a^3*b^3 - 483*A*a^2*b^4)*c^2 - 420*(150*B*a^2*b^5 - 97*A*a
*b^6)*c)*x^8 - 2*(1890*B*a^2*b^6 - 1155*A*a*b^7 - 64*(630*B*a^5 - 689*A*a^4*b)*c
^3 + 144*(398*B*a^4*b^2 - 293*A*a^3*b^3)*c^2 - 84*(226*B*a^3*b^4 - 147*A*a^2*b^5
)*c)*x^7 + 8*(378*B*a^3*b^5 - 231*A*a^2*b^6 + 4096*A*a^5*c^3 + 48*(174*B*a^5*b -
133*A*a^4*b^2)*c^2 - 24*(142*B*a^4*b^3 - 93*A*a^3*b^4)*c)*x^6 - 16*(162*B*a^4*b
^4 - 99*A*a^3*b^5 - 48*(826*B*a^6 + 41*A*a^5*b)*c^2 - 8*(162*B*a^5*b^2 - 107*A*a
^4*b^3)*c)*x^5 + 128*(18*B*a^5*b^3 - 11*A*a^4*b^4 + 3840*A*a^6*c^2 + 12*(614*B*a
^6*b + 7*A*a^5*b^2)*c)*x^4 + 256*(1458*B*a^6*b^2 + 5*A*a^5*b^3 + 12*(238*B*a^7 +
251*A*a^6*b)*c)*x^3 + 1024*(594*B*a^7*b + 309*A*a^6*b^2 + 608*A*a^7*c)*x^2 + 14
336*(18*B*a^8 + 37*A*a^7*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^6*x^9
)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{x^{10}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(5/2)/x**10,x)

[Out]

Integral((A + B*x)*(a + b*x + c*x**2)**(5/2)/x**10, x)

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GIAC/XCAS [A]  time = 0.340452, 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^10,x, algorithm="giac")

[Out]

Done