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

Optimal. Leaf size=172 \[ \frac{5 B c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}+\frac{5 B c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6} \]

[Out]

(5*B*c^3*Sqrt[a + c*x^2])/(128*a*x^2) + (5*B*c^2*(a + c*x^2)^(3/2))/(192*a*x^4)
+ (B*c*(a + c*x^2)^(5/2))/(48*a*x^6) - (A*(a + c*x^2)^(7/2))/(9*a*x^9) - (B*(a +
c*x^2)^(7/2))/(8*a*x^8) + (2*A*c*(a + c*x^2)^(7/2))/(63*a^2*x^7) + (5*B*c^4*Arc
Tanh[Sqrt[a + c*x^2]/Sqrt[a]])/(128*a^(3/2))

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Rubi [A]  time = 0.344856, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{5 B c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}-\frac{A \left (a+c x^2\right )^{7/2}}{9 a x^9}+\frac{5 B c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}-\frac{B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{B c \left (a+c x^2\right )^{5/2}}{48 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

(5*B*c^3*Sqrt[a + c*x^2])/(128*a*x^2) + (5*B*c^2*(a + c*x^2)^(3/2))/(192*a*x^4)
+ (B*c*(a + c*x^2)^(5/2))/(48*a*x^6) - (A*(a + c*x^2)^(7/2))/(9*a*x^9) - (B*(a +
c*x^2)^(7/2))/(8*a*x^8) + (2*A*c*(a + c*x^2)^(7/2))/(63*a^2*x^7) + (5*B*c^4*Arc
Tanh[Sqrt[a + c*x^2]/Sqrt[a]])/(128*a^(3/2))

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Rubi in Sympy [A]  time = 36.0894, size = 158, normalized size = 0.92 \[ - \frac{A \left (a + c x^{2}\right )^{\frac{7}{2}}}{9 a x^{9}} + \frac{2 A c \left (a + c x^{2}\right )^{\frac{7}{2}}}{63 a^{2} x^{7}} + \frac{5 B c^{3} \sqrt{a + c x^{2}}}{128 a x^{2}} + \frac{5 B c^{2} \left (a + c x^{2}\right )^{\frac{3}{2}}}{192 a x^{4}} + \frac{B c \left (a + c x^{2}\right )^{\frac{5}{2}}}{48 a x^{6}} - \frac{B \left (a + c x^{2}\right )^{\frac{7}{2}}}{8 a x^{8}} + \frac{5 B c^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + c*x**2)**(7/2)/(9*a*x**9) + 2*A*c*(a + c*x**2)**(7/2)/(63*a**2*x**7) + 5
*B*c**3*sqrt(a + c*x**2)/(128*a*x**2) + 5*B*c**2*(a + c*x**2)**(3/2)/(192*a*x**4
) + B*c*(a + c*x**2)**(5/2)/(48*a*x**6) - B*(a + c*x**2)**(7/2)/(8*a*x**8) + 5*B
*c**4*atanh(sqrt(a + c*x**2)/sqrt(a))/(128*a**(3/2))

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Mathematica [A]  time = 0.536076, size = 142, normalized size = 0.83 \[ -\frac{\frac{\sqrt{a+c x^2} \left (112 a^4 (8 A+9 B x)+8 a^3 c x^2 (304 A+357 B x)+6 a^2 c^2 x^4 (320 A+413 B x)+a c^3 x^6 (128 A+315 B x)-256 A c^4 x^8\right )}{x^9}-315 \sqrt{a} B c^4 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+315 \sqrt{a} B c^4 \log (x)}{8064 a^2} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[a + c*x^2]*(-256*A*c^4*x^8 + 112*a^4*(8*A + 9*B*x) + a*c^3*x^6*(128*A +
315*B*x) + 8*a^3*c*x^2*(304*A + 357*B*x) + 6*a^2*c^2*x^4*(320*A + 413*B*x)))/x^9
+ 315*Sqrt[a]*B*c^4*Log[x] - 315*Sqrt[a]*B*c^4*Log[a + Sqrt[a]*Sqrt[a + c*x^2]]
)/(8064*a^2)

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Maple [A]  time = 0.04, size = 204, normalized size = 1.2 \[ -{\frac{A}{9\,a{x}^{9}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{2\,Ac}{63\,{a}^{2}{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B}{8\,a{x}^{8}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Bc}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{c}^{2}}{192\,{a}^{3}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{B{c}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B{c}^{4}}{128\,{a}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,B{c}^{4}}{384\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,B{c}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,B{c}^{4}}{128\,{a}^{2}}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*A*(c*x^2+a)^(7/2)/a/x^9+2/63*A*c*(c*x^2+a)^(7/2)/a^2/x^7-1/8*B*(c*x^2+a)^(7
/2)/a/x^8+1/48*B/a^2*c/x^6*(c*x^2+a)^(7/2)+1/192*B/a^3*c^2/x^4*(c*x^2+a)^(7/2)+1
/128*B/a^4*c^3/x^2*(c*x^2+a)^(7/2)-1/128*B/a^4*c^4*(c*x^2+a)^(5/2)-5/384*B/a^3*c
^4*(c*x^2+a)^(3/2)+5/128*B/a^(3/2)*c^4*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-5/1
28*B/a^2*c^4*(c*x^2+a)^(1/2)

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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 + a)^(5/2)*(B*x + A)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.512256, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, B a c^{4} x^{9} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{16128 \, a^{\frac{5}{2}} x^{9}}, \frac{315 \, B a c^{4} x^{9} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{8064 \, \sqrt{-a} a^{2} x^{9}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^10,x, algorithm="fricas")

[Out]

[1/16128*(315*B*a*c^4*x^9*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2
) + 2*(256*A*c^4*x^8 - 315*B*a*c^3*x^7 - 128*A*a*c^3*x^6 - 2478*B*a^2*c^2*x^5 -
1920*A*a^2*c^2*x^4 - 2856*B*a^3*c*x^3 - 2432*A*a^3*c*x^2 - 1008*B*a^4*x - 896*A*
a^4)*sqrt(c*x^2 + a)*sqrt(a))/(a^(5/2)*x^9), 1/8064*(315*B*a*c^4*x^9*arctan(sqrt
(-a)/sqrt(c*x^2 + a)) + (256*A*c^4*x^8 - 315*B*a*c^3*x^7 - 128*A*a*c^3*x^6 - 247
8*B*a^2*c^2*x^5 - 1920*A*a^2*c^2*x^4 - 2856*B*a^3*c*x^3 - 2432*A*a^3*c*x^2 - 100
8*B*a^4*x - 896*A*a^4)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^9)]

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Sympy [A]  time = 76.1941, size = 1202, normalized size = 6.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-35*A*a**9*c**(19/2)*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x
**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 110*A*a**8*c**(21/2)*x**2*
sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11
*x**12 + 315*a**4*c**12*x**14) - 114*A*a**7*c**(23/2)*x**4*sqrt(a/(c*x**2) + 1)/
(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**
12*x**14) - 40*A*a**6*c**(25/2)*x**6*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 +
945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 30*A*a**6*
c**(11/2)*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a*
*3*c**6*x**10) + 5*A*a**5*c**(27/2)*x**8*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**
8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 66*A*a
**5*c**(13/2)*x**2*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8
+ 105*a**3*c**6*x**10) + 30*A*a**4*c**(29/2)*x**10*sqrt(a/(c*x**2) + 1)/(315*a*
*7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**1
4) - 34*A*a**4*c**(15/2)*x**4*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**
4*c**5*x**8 + 105*a**3*c**6*x**10) + 40*A*a**3*c**(31/2)*x**12*sqrt(a/(c*x**2) +
1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4
*c**12*x**14) - 6*A*a**3*c**(17/2)*x**6*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6
+ 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) + 16*A*a**2*c**(33/2)*x**14*sqrt(a/
(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12
+ 315*a**4*c**12*x**14) - 24*A*a**2*c**(19/2)*x**8*sqrt(a/(c*x**2) + 1)/(105*a**
5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 16*A*a*c**(21/2)*x**10
*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x
**10) - A*c**(5/2)*sqrt(a/(c*x**2) + 1)/(5*x**4) - A*c**(7/2)*sqrt(a/(c*x**2) +
1)/(15*a*x**2) + 2*A*c**(9/2)*sqrt(a/(c*x**2) + 1)/(15*a**2) - B*a**3/(8*sqrt(c)
*x**9*sqrt(a/(c*x**2) + 1)) - 23*B*a**2*sqrt(c)/(48*x**7*sqrt(a/(c*x**2) + 1)) -
127*B*a*c**(3/2)/(192*x**5*sqrt(a/(c*x**2) + 1)) - 133*B*c**(5/2)/(384*x**3*sqr
t(a/(c*x**2) + 1)) - 5*B*c**(7/2)/(128*a*x*sqrt(a/(c*x**2) + 1)) + 5*B*c**4*asin
h(sqrt(a)/(sqrt(c)*x))/(128*a**(3/2))

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GIAC/XCAS [A]  time = 0.285402, size = 663, normalized size = 3.85 \[ -\frac{5 \, B c^{4} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a} + \frac{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{17} B c^{4} + 8022 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{15} B a c^{4} + 16128 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{14} A a c^{\frac{9}{2}} + 10458 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} B a^{2} c^{4} + 26880 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{12} A a^{2} c^{\frac{9}{2}} + 18270 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} B a^{3} c^{4} + 80640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} A a^{3} c^{\frac{9}{2}} + 48384 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A a^{4} c^{\frac{9}{2}} - 18270 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a^{5} c^{4} + 48384 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} A a^{5} c^{\frac{9}{2}} - 10458 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B a^{6} c^{4} + 6912 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{6} c^{\frac{9}{2}} - 8022 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{7} c^{4} + 2304 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{7} c^{\frac{9}{2}} - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{8} c^{4} - 256 \, A a^{8} c^{\frac{9}{2}}}{4032 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{9} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^10,x, algorithm="giac")

[Out]

-5/64*B*c^4*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/403
2*(315*(sqrt(c)*x - sqrt(c*x^2 + a))^17*B*c^4 + 8022*(sqrt(c)*x - sqrt(c*x^2 + a
))^15*B*a*c^4 + 16128*(sqrt(c)*x - sqrt(c*x^2 + a))^14*A*a*c^(9/2) + 10458*(sqrt
(c)*x - sqrt(c*x^2 + a))^13*B*a^2*c^4 + 26880*(sqrt(c)*x - sqrt(c*x^2 + a))^12*A
*a^2*c^(9/2) + 18270*(sqrt(c)*x - sqrt(c*x^2 + a))^11*B*a^3*c^4 + 80640*(sqrt(c)
*x - sqrt(c*x^2 + a))^10*A*a^3*c^(9/2) + 48384*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A
*a^4*c^(9/2) - 18270*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*a^5*c^4 + 48384*(sqrt(c)*
x - sqrt(c*x^2 + a))^6*A*a^5*c^(9/2) - 10458*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*a
^6*c^4 + 6912*(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^6*c^(9/2) - 8022*(sqrt(c)*x -
sqrt(c*x^2 + a))^3*B*a^7*c^4 + 2304*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a^7*c^(9/2
) - 315*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^8*c^4 - 256*A*a^8*c^(9/2))/(((sqrt(c)*
x - sqrt(c*x^2 + a))^2 - a)^9*a)