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

Optimal. Leaf size=149 \[ \frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]

[Out]

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

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Rubi [A]  time = 0.256817, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.7877, size = 134, normalized size = 0.9 \[ \frac{5 A c^{3} \sqrt{a + c x^{2}}}{128 a x^{2}} + \frac{5 A c^{2} \left (a + c x^{2}\right )^{\frac{3}{2}}}{192 a x^{4}} + \frac{A c \left (a + c x^{2}\right )^{\frac{5}{2}}}{48 a x^{6}} - \frac{A \left (a + c x^{2}\right )^{\frac{7}{2}}}{8 a x^{8}} + \frac{5 A c^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{3}{2}}} - \frac{B \left (a + c x^{2}\right )^{\frac{7}{2}}}{7 a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.208311, size = 135, normalized size = 0.91 \[ \frac{-\sqrt{a} \sqrt{a+c x^2} \left (48 a^3 (7 A+8 B x)+8 a^2 c x^2 (119 A+144 B x)+2 a c^2 x^4 (413 A+576 B x)+3 c^3 x^6 (35 A+128 B x)\right )+105 A c^4 x^8 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-105 A c^4 x^8 \log (x)}{2688 a^{3/2} x^8} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[a]*Sqrt[a + c*x^2]*(48*a^3*(7*A + 8*B*x) + 3*c^3*x^6*(35*A + 128*B*x) +
8*a^2*c*x^2*(119*A + 144*B*x) + 2*a*c^2*x^4*(413*A + 576*B*x))) - 105*A*c^4*x^8*
Log[x] + 105*A*c^4*x^8*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(2688*a^(3/2)*x^8)

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Maple [A]  time = 0.031, size = 185, normalized size = 1.2 \[ -{\frac{A}{8\,a{x}^{8}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ac}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{2}}{192\,{a}^{3}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{c}^{4}}{128\,{a}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,A{c}^{4}}{384\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{c}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,A{c}^{4}}{128\,{a}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{B}{7\,a{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.42956, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, A c^{4} x^{8} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (384 \, B c^{3} x^{7} + 105 \, A c^{3} x^{6} + 1152 \, B a c^{2} x^{5} + 826 \, A a c^{2} x^{4} + 1152 \, B a^{2} c x^{3} + 952 \, A a^{2} c x^{2} + 384 \, B a^{3} x + 336 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{5376 \, a^{\frac{3}{2}} x^{8}}, \frac{105 \, A c^{4} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (384 \, B c^{3} x^{7} + 105 \, A c^{3} x^{6} + 1152 \, B a c^{2} x^{5} + 826 \, A a c^{2} x^{4} + 1152 \, B a^{2} c x^{3} + 952 \, A a^{2} c x^{2} + 384 \, B a^{3} x + 336 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{2688 \, \sqrt{-a} a x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/5376*(105*A*c^4*x^8*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2) -
2*(384*B*c^3*x^7 + 105*A*c^3*x^6 + 1152*B*a*c^2*x^5 + 826*A*a*c^2*x^4 + 1152*B*
a^2*c*x^3 + 952*A*a^2*c*x^2 + 384*B*a^3*x + 336*A*a^3)*sqrt(c*x^2 + a)*sqrt(a))/
(a^(3/2)*x^8), 1/2688*(105*A*c^4*x^8*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (384*B*c
^3*x^7 + 105*A*c^3*x^6 + 1152*B*a*c^2*x^5 + 826*A*a*c^2*x^4 + 1152*B*a^2*c*x^3 +
952*A*a^2*c*x^2 + 384*B*a^3*x + 336*A*a^3)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*
a*x^8)]

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Sympy [A]  time = 71.7417, size = 609, normalized size = 4.09 \[ - \frac{A a^{3}}{8 \sqrt{c} x^{9} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{23 A a^{2} \sqrt{c}}{48 x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{127 A a c^{\frac{3}{2}}}{192 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{133 A c^{\frac{5}{2}}}{384 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 A c^{\frac{7}{2}}}{128 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{5 A c^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{128 a^{\frac{3}{2}}} - \frac{15 B a^{7} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{33 B a^{6} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{17 B a^{5} c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{3 B a^{4} c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{12 B a^{3} c^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{8 B a^{2} c^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{2 B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{7 B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{B c^{\frac{7}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**3/(8*sqrt(c)*x**9*sqrt(a/(c*x**2) + 1)) - 23*A*a**2*sqrt(c)/(48*x**7*sqrt(
a/(c*x**2) + 1)) - 127*A*a*c**(3/2)/(192*x**5*sqrt(a/(c*x**2) + 1)) - 133*A*c**(
5/2)/(384*x**3*sqrt(a/(c*x**2) + 1)) - 5*A*c**(7/2)/(128*a*x*sqrt(a/(c*x**2) + 1
)) + 5*A*c**4*asinh(sqrt(a)/(sqrt(c)*x))/(128*a**(3/2)) - 15*B*a**7*c**(9/2)*sqr
t(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
) - 33*B*a**6*c**(11/2)*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) - 17*B*a**5*c**(13/2)*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) - 3*B*a**4*c**
(15/2)*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) - 12*B*a**3*c**(17/2)*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) - 8*B*a**2*c**(19/2)*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)
- 2*B*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 7*B*c**(5/2)*sqrt(a/(c*x**2) +
1)/(15*x**2) - B*c**(7/2)*sqrt(a/(c*x**2) + 1)/(15*a)

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GIAC/XCAS [A]  time = 0.289388, size = 663, normalized size = 4.45 \[ -\frac{5 \, A c^{4} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a} + \frac{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{15} A c^{4} + 2688 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{14} B a c^{\frac{7}{2}} + 2779 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} A a c^{4} - 2688 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{12} B a^{2} c^{\frac{7}{2}} + 6265 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} A a^{2} c^{4} + 13440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} B a^{3} c^{\frac{7}{2}} + 12355 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} A a^{3} c^{4} - 13440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} B a^{4} c^{\frac{7}{2}} + 12355 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a^{4} c^{4} + 8064 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{5} c^{\frac{7}{2}} + 6265 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{5} c^{4} - 8064 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{6} c^{\frac{7}{2}} + 2779 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{6} c^{4} + 384 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{7} c^{\frac{7}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{7} c^{4} - 384 \, B a^{8} c^{\frac{7}{2}}}{1344 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{8} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/64*A*c^4*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/134
4*(105*(sqrt(c)*x - sqrt(c*x^2 + a))^15*A*c^4 + 2688*(sqrt(c)*x - sqrt(c*x^2 + a
))^14*B*a*c^(7/2) + 2779*(sqrt(c)*x - sqrt(c*x^2 + a))^13*A*a*c^4 - 2688*(sqrt(c
)*x - sqrt(c*x^2 + a))^12*B*a^2*c^(7/2) + 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^11*
A*a^2*c^4 + 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^10*B*a^3*c^(7/2) + 12355*(sqrt(c
)*x - sqrt(c*x^2 + a))^9*A*a^3*c^4 - 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^8*B*a^4
*c^(7/2) + 12355*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*a^4*c^4 + 8064*(sqrt(c)*x - s
qrt(c*x^2 + a))^6*B*a^5*c^(7/2) + 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*a^5*c^4
- 8064*(sqrt(c)*x - sqrt(c*x^2 + a))^4*B*a^6*c^(7/2) + 2779*(sqrt(c)*x - sqrt(c
*x^2 + a))^3*A*a^6*c^4 + 384*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a^7*c^(7/2) + 105
*(sqrt(c)*x - sqrt(c*x^2 + a))*A*a^7*c^4 - 384*B*a^8*c^(7/2))/(((sqrt(c)*x - sqr
t(c*x^2 + a))^2 - a)^8*a)