3.338 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=147 \[ \frac{B c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac{A \left (a+c x^2\right )^{5/2}}{7 a x^7}+\frac{B c^2 \sqrt{a+c x^2}}{16 a x^2}-\frac{B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac{B c \left (a+c x^2\right )^{3/2}}{24 a x^4} \]

[Out]

(B*c^2*Sqrt[a + c*x^2])/(16*a*x^2) + (B*c*(a + c*x^2)^(3/2))/(24*a*x^4) - (A*(a
+ c*x^2)^(5/2))/(7*a*x^7) - (B*(a + c*x^2)^(5/2))/(6*a*x^6) + (2*A*c*(a + c*x^2)
^(5/2))/(35*a^2*x^5) + (B*c^3*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(16*a^(3/2))

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Rubi [A]  time = 0.303497, antiderivative size = 147, 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 \[ \frac{B c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{2 A c \left (a+c x^2\right )^{5/2}}{35 a^2 x^5}-\frac{A \left (a+c x^2\right )^{5/2}}{7 a x^7}+\frac{B c^2 \sqrt{a+c x^2}}{16 a x^2}-\frac{B \left (a+c x^2\right )^{5/2}}{6 a x^6}+\frac{B c \left (a+c x^2\right )^{3/2}}{24 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(3/2))/x^8,x]

[Out]

(B*c^2*Sqrt[a + c*x^2])/(16*a*x^2) + (B*c*(a + c*x^2)^(3/2))/(24*a*x^4) - (A*(a
+ c*x^2)^(5/2))/(7*a*x^7) - (B*(a + c*x^2)^(5/2))/(6*a*x^6) + (2*A*c*(a + c*x^2)
^(5/2))/(35*a^2*x^5) + (B*c^3*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(16*a^(3/2))

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Rubi in Sympy [A]  time = 31.847, size = 131, normalized size = 0.89 \[ - \frac{A \left (a + c x^{2}\right )^{\frac{5}{2}}}{7 a x^{7}} + \frac{2 A c \left (a + c x^{2}\right )^{\frac{5}{2}}}{35 a^{2} x^{5}} + \frac{B c^{2} \sqrt{a + c x^{2}}}{16 a x^{2}} + \frac{B c \left (a + c x^{2}\right )^{\frac{3}{2}}}{24 a x^{4}} - \frac{B \left (a + c x^{2}\right )^{\frac{5}{2}}}{6 a x^{6}} + \frac{B c^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/x**8,x)

[Out]

-A*(a + c*x**2)**(5/2)/(7*a*x**7) + 2*A*c*(a + c*x**2)**(5/2)/(35*a**2*x**5) + B
*c**2*sqrt(a + c*x**2)/(16*a*x**2) + B*c*(a + c*x**2)**(3/2)/(24*a*x**4) - B*(a
+ c*x**2)**(5/2)/(6*a*x**6) + B*c**3*atanh(sqrt(a + c*x**2)/sqrt(a))/(16*a**(3/2
))

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Mathematica [A]  time = 0.379884, size = 124, normalized size = 0.84 \[ -\frac{\frac{\sqrt{a+c x^2} \left (40 a^3 (6 A+7 B x)+2 a^2 c x^2 (192 A+245 B x)+3 a c^2 x^4 (16 A+35 B x)-96 A c^3 x^6\right )}{x^7}-105 \sqrt{a} B c^3 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+105 \sqrt{a} B c^3 \log (x)}{1680 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(3/2))/x^8,x]

[Out]

-((Sqrt[a + c*x^2]*(-96*A*c^3*x^6 + 40*a^3*(6*A + 7*B*x) + 3*a*c^2*x^4*(16*A + 3
5*B*x) + 2*a^2*c*x^2*(192*A + 245*B*x)))/x^7 + 105*Sqrt[a]*B*c^3*Log[x] - 105*Sq
rt[a]*B*c^3*Log[a + Sqrt[a]*Sqrt[a + c*x^2]])/(1680*a^2)

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Maple [A]  time = 0.022, size = 165, normalized size = 1.1 \[ -{\frac{A}{7\,a{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,Ac}{35\,{a}^{2}{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B}{6\,a{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bc}{24\,{a}^{2}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{c}^{2}}{48\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B{c}^{3}}{48\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{c}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B{c}^{3}}{16\,{a}^{2}}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(3/2)/x^8,x)

[Out]

-1/7*A*(c*x^2+a)^(5/2)/a/x^7+2/35*A*c*(c*x^2+a)^(5/2)/a^2/x^5-1/6*B*(c*x^2+a)^(5
/2)/a/x^6+1/24*B/a^2*c/x^4*(c*x^2+a)^(5/2)+1/48*B/a^3*c^2/x^2*(c*x^2+a)^(5/2)-1/
48*B/a^3*c^3*(c*x^2+a)^(3/2)+1/16*B/a^(3/2)*c^3*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2
))/x)-1/16*B/a^2*c^3*(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)^(3/2)*(B*x + A)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.367713, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a c^{3} x^{7} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (96 \, A c^{3} x^{6} - 105 \, B a c^{2} x^{5} - 48 \, A a c^{2} x^{4} - 490 \, B a^{2} c x^{3} - 384 \, A a^{2} c x^{2} - 280 \, B a^{3} x - 240 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{3360 \, a^{\frac{5}{2}} x^{7}}, \frac{105 \, B a c^{3} x^{7} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (96 \, A c^{3} x^{6} - 105 \, B a c^{2} x^{5} - 48 \, A a c^{2} x^{4} - 490 \, B a^{2} c x^{3} - 384 \, A a^{2} c x^{2} - 280 \, B a^{3} x - 240 \, A a^{3}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{1680 \, \sqrt{-a} a^{2} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^8,x, algorithm="fricas")

[Out]

[1/3360*(105*B*a*c^3*x^7*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2)
+ 2*(96*A*c^3*x^6 - 105*B*a*c^2*x^5 - 48*A*a*c^2*x^4 - 490*B*a^2*c*x^3 - 384*A*
a^2*c*x^2 - 280*B*a^3*x - 240*A*a^3)*sqrt(c*x^2 + a)*sqrt(a))/(a^(5/2)*x^7), 1/1
680*(105*B*a*c^3*x^7*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (96*A*c^3*x^6 - 105*B*a*
c^2*x^5 - 48*A*a*c^2*x^4 - 490*B*a^2*c*x^3 - 384*A*a^2*c*x^2 - 280*B*a^3*x - 240
*A*a^3)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^7)]

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Sympy [A]  time = 39.2049, size = 575, normalized size = 3.91 \[ - \frac{15 A a^{6} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{33 A a^{5} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{17 A a^{4} c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{3 A a^{3} c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{12 A a^{2} c^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{8 A a c^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac{A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a x^{2}} + \frac{2 A c^{\frac{7}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{2}} - \frac{B a^{2}}{6 \sqrt{c} x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{11 B a \sqrt{c}}{24 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{17 B c^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B c^{\frac{5}{2}}}{16 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{16 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(3/2)/x**8,x)

[Out]

-15*A*a**6*c**(9/2)*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) - 33*A*a**5*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*A*a**4*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*A*a**3*c**(15/2)*x**6*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 21
0*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 12*A*a**2*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*A*a*
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) - A*c**(3/2)*sqrt(a/(c*x**2) + 1)/(5*x**4) - A*c**(5/2)*sqr
t(a/(c*x**2) + 1)/(15*a*x**2) + 2*A*c**(7/2)*sqrt(a/(c*x**2) + 1)/(15*a**2) - B*
a**2/(6*sqrt(c)*x**7*sqrt(a/(c*x**2) + 1)) - 11*B*a*sqrt(c)/(24*x**5*sqrt(a/(c*x
**2) + 1)) - 17*B*c**(3/2)/(48*x**3*sqrt(a/(c*x**2) + 1)) - B*c**(5/2)/(16*a*x*s
qrt(a/(c*x**2) + 1)) + B*c**3*asinh(sqrt(a)/(sqrt(c)*x))/(16*a**(3/2))

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GIAC/XCAS [A]  time = 0.281701, size = 512, normalized size = 3.48 \[ -\frac{B c^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a} + \frac{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} B c^{3} + 1540 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} B a c^{3} + 3360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} A a c^{\frac{7}{2}} + 1085 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B a^{2} c^{3} + 3360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A a^{2} c^{\frac{7}{2}} + 6720 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} A a^{3} c^{\frac{7}{2}} - 1085 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} B a^{4} c^{3} + 1344 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{4} c^{\frac{7}{2}} - 1540 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{5} c^{3} + 672 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{5} c^{\frac{7}{2}} - 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{6} c^{3} - 96 \, A a^{6} c^{\frac{7}{2}}}{840 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{7} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^8,x, algorithm="giac")

[Out]

-1/8*B*c^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/840*
(105*(sqrt(c)*x - sqrt(c*x^2 + a))^13*B*c^3 + 1540*(sqrt(c)*x - sqrt(c*x^2 + a))
^11*B*a*c^3 + 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^10*A*a*c^(7/2) + 1085*(sqrt(c)*
x - sqrt(c*x^2 + a))^9*B*a^2*c^3 + 3360*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A*a^2*c^
(7/2) + 6720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*A*a^3*c^(7/2) - 1085*(sqrt(c)*x - s
qrt(c*x^2 + a))^5*B*a^4*c^3 + 1344*(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^4*c^(7/2)
- 1540*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^5*c^3 + 672*(sqrt(c)*x - sqrt(c*x^2
+ a))^2*A*a^5*c^(7/2) - 105*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^6*c^3 - 96*A*a^6*c
^(7/2))/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^7*a)