3.350 \(\int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=140 \[ -\frac{c^2 \sqrt{a+c x^2} (5 A+16 B x)}{16 x^2}-\frac{\left (a+c x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}-\frac{c \left (a+c x^2\right )^{3/2} (5 A+8 B x)}{24 x^4}-\frac{5 A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}+B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

[Out]

-(c^2*(5*A + 16*B*x)*Sqrt[a + c*x^2])/(16*x^2) - (c*(5*A + 8*B*x)*(a + c*x^2)^(3
/2))/(24*x^4) - ((5*A + 6*B*x)*(a + c*x^2)^(5/2))/(30*x^6) + B*c^(5/2)*ArcTanh[(
Sqrt[c]*x)/Sqrt[a + c*x^2]] - (5*A*c^3*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(16*Sqr
t[a])

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Rubi [A]  time = 0.355361, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{c^2 \sqrt{a+c x^2} (5 A+16 B x)}{16 x^2}-\frac{\left (a+c x^2\right )^{5/2} (5 A+6 B x)}{30 x^6}-\frac{c \left (a+c x^2\right )^{3/2} (5 A+8 B x)}{24 x^4}-\frac{5 A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}+B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(5/2))/x^7,x]

[Out]

-(c^2*(5*A + 16*B*x)*Sqrt[a + c*x^2])/(16*x^2) - (c*(5*A + 8*B*x)*(a + c*x^2)^(3
/2))/(24*x^4) - ((5*A + 6*B*x)*(a + c*x^2)^(5/2))/(30*x^6) + B*c^(5/2)*ArcTanh[(
Sqrt[c]*x)/Sqrt[a + c*x^2]] - (5*A*c^3*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(16*Sqr
t[a])

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Rubi in Sympy [A]  time = 48.4669, size = 131, normalized size = 0.94 \[ - \frac{5 A c^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{16 \sqrt{a}} + B c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )} - \frac{c^{2} \left (60 A + 192 B x\right ) \sqrt{a + c x^{2}}}{192 x^{2}} - \frac{c \left (30 A + 48 B x\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{144 x^{4}} - \frac{\left (5 A + 6 B x\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{30 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/x**7,x)

[Out]

-5*A*c**3*atanh(sqrt(a + c*x**2)/sqrt(a))/(16*sqrt(a)) + B*c**(5/2)*atanh(sqrt(c
)*x/sqrt(a + c*x**2)) - c**2*(60*A + 192*B*x)*sqrt(a + c*x**2)/(192*x**2) - c*(3
0*A + 48*B*x)*(a + c*x**2)**(3/2)/(144*x**4) - (5*A + 6*B*x)*(a + c*x**2)**(5/2)
/(30*x**6)

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Mathematica [A]  time = 0.524852, size = 140, normalized size = 1. \[ -\frac{\sqrt{a+c x^2} \left (8 a^2 (5 A+6 B x)+2 a c x^2 (65 A+88 B x)+c^2 x^4 (165 A+368 B x)\right )}{240 x^6}-\frac{5 A c^3 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{16 \sqrt{a}}+\frac{5 A c^3 \log (x)}{16 \sqrt{a}}+B c^{5/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^7,x]

[Out]

-(Sqrt[a + c*x^2]*(8*a^2*(5*A + 6*B*x) + 2*a*c*x^2*(65*A + 88*B*x) + c^2*x^4*(16
5*A + 368*B*x)))/(240*x^6) + (5*A*c^3*Log[x])/(16*Sqrt[a]) - (5*A*c^3*Log[a + Sq
rt[a]*Sqrt[a + c*x^2]])/(16*Sqrt[a]) + B*c^(5/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^
2]]

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Maple [B]  time = 0.02, size = 281, normalized size = 2. \[ -{\frac{A}{6\,a{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Ac}{24\,{a}^{2}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{c}^{2}}{16\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{3}}{16\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,A{c}^{3}}{48\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{c}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{5\,A{c}^{3}}{16\,a}\sqrt{c{x}^{2}+a}}-{\frac{B}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Bc}{15\,{a}^{2}{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,B{c}^{2}}{15\,{a}^{3}x} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,B{c}^{3}x}{15\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,B{c}^{3}x}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{c}^{3}x}{a}\sqrt{c{x}^{2}+a}}+B{c}^{{\frac{5}{2}}}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(5/2)/x^7,x)

[Out]

-1/6*A/a/x^6*(c*x^2+a)^(7/2)-1/24*A/a^2*c/x^4*(c*x^2+a)^(7/2)-1/16*A/a^3*c^2/x^2
*(c*x^2+a)^(7/2)+1/16*A/a^3*c^3*(c*x^2+a)^(5/2)+5/48*A/a^2*c^3*(c*x^2+a)^(3/2)-5
/16*A/a^(1/2)*c^3*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+5/16*A/a*c^3*(c*x^2+a)^(
1/2)-1/5*B/a/x^5*(c*x^2+a)^(7/2)-2/15*B/a^2*c/x^3*(c*x^2+a)^(7/2)-8/15*B/a^3*c^2
/x*(c*x^2+a)^(7/2)+8/15*B/a^3*c^3*x*(c*x^2+a)^(5/2)+2/3*B/a^2*c^3*x*(c*x^2+a)^(3
/2)+B/a*c^3*x*(c*x^2+a)^(1/2)+B*c^(5/2)*ln(c^(1/2)*x+(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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.352274, size = 1, normalized size = 0.01 \[ \left [\frac{240 \, B \sqrt{a} c^{\frac{5}{2}} x^{6} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 75 \, A c^{3} x^{6} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{480 \, \sqrt{a} x^{6}}, \frac{480 \, B \sqrt{a} \sqrt{-c} c^{2} x^{6} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) + 75 \, A c^{3} x^{6} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{480 \, \sqrt{a} x^{6}}, -\frac{75 \, A c^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) - 120 \, B \sqrt{-a} c^{\frac{5}{2}} x^{6} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) +{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{240 \, \sqrt{-a} x^{6}}, \frac{240 \, B \sqrt{-a} \sqrt{-c} c^{2} x^{6} \arctan \left (\frac{c x}{\sqrt{c x^{2} + a} \sqrt{-c}}\right ) - 75 \, A c^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (368 \, B c^{2} x^{5} + 165 \, A c^{2} x^{4} + 176 \, B a c x^{3} + 130 \, A a c x^{2} + 48 \, B a^{2} x + 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{240 \, \sqrt{-a} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

[1/480*(240*B*sqrt(a)*c^(5/2)*x^6*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a
) + 75*A*c^3*x^6*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) - 2*(36
8*B*c^2*x^5 + 165*A*c^2*x^4 + 176*B*a*c*x^3 + 130*A*a*c*x^2 + 48*B*a^2*x + 40*A*
a^2)*sqrt(c*x^2 + a)*sqrt(a))/(sqrt(a)*x^6), 1/480*(480*B*sqrt(a)*sqrt(-c)*c^2*x
^6*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) + 75*A*c^3*x^6*log(-((c*x^2 + 2*a)*sqr
t(a) - 2*sqrt(c*x^2 + a)*a)/x^2) - 2*(368*B*c^2*x^5 + 165*A*c^2*x^4 + 176*B*a*c*
x^3 + 130*A*a*c*x^2 + 48*B*a^2*x + 40*A*a^2)*sqrt(c*x^2 + a)*sqrt(a))/(sqrt(a)*x
^6), -1/240*(75*A*c^3*x^6*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - 120*B*sqrt(-a)*c^(5
/2)*x^6*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (368*B*c^2*x^5 + 165*A
*c^2*x^4 + 176*B*a*c*x^3 + 130*A*a*c*x^2 + 48*B*a^2*x + 40*A*a^2)*sqrt(c*x^2 + a
)*sqrt(-a))/(sqrt(-a)*x^6), 1/240*(240*B*sqrt(-a)*sqrt(-c)*c^2*x^6*arctan(c*x/(s
qrt(c*x^2 + a)*sqrt(-c))) - 75*A*c^3*x^6*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (368
*B*c^2*x^5 + 165*A*c^2*x^4 + 176*B*a*c*x^3 + 130*A*a*c*x^2 + 48*B*a^2*x + 40*A*a
^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*x^6)]

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Sympy [A]  time = 47.1005, size = 299, normalized size = 2.14 \[ - \frac{A a^{3}}{6 \sqrt{c} x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{17 A a^{2} \sqrt{c}}{24 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{35 A a c^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{3 A c^{\frac{5}{2}}}{16 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 A c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{16 \sqrt{a}} - \frac{B \sqrt{a} c^{2}}{x \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{B a^{2} \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{11 B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{8 B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15} + B c^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )} - \frac{B c^{3} x}{\sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(5/2)/x**7,x)

[Out]

-A*a**3/(6*sqrt(c)*x**7*sqrt(a/(c*x**2) + 1)) - 17*A*a**2*sqrt(c)/(24*x**5*sqrt(
a/(c*x**2) + 1)) - 35*A*a*c**(3/2)/(48*x**3*sqrt(a/(c*x**2) + 1)) - A*c**(5/2)*s
qrt(a/(c*x**2) + 1)/(2*x) - 3*A*c**(5/2)/(16*x*sqrt(a/(c*x**2) + 1)) - 5*A*c**3*
asinh(sqrt(a)/(sqrt(c)*x))/(16*sqrt(a)) - B*sqrt(a)*c**2/(x*sqrt(1 + c*x**2/a))
- B*a**2*sqrt(c)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 11*B*a*c**(3/2)*sqrt(a/(c*x**2)
+ 1)/(15*x**2) - 8*B*c**(5/2)*sqrt(a/(c*x**2) + 1)/15 + B*c**(5/2)*asinh(sqrt(c
)*x/sqrt(a)) - B*c**3*x/(sqrt(a)*sqrt(1 + c*x**2/a))

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GIAC/XCAS [A]  time = 0.295582, size = 536, normalized size = 3.83 \[ \frac{5 \, A c^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a}} - B c^{\frac{5}{2}}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} A c^{3} + 720 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} B a c^{\frac{5}{2}} + 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} A a c^{3} - 2160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} B a^{2} c^{\frac{5}{2}} + 450 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a^{2} c^{3} + 3680 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{3} c^{\frac{5}{2}} + 450 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{3} c^{3} - 3360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{4} c^{\frac{5}{2}} + 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{4} c^{3} + 1488 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{5} c^{\frac{5}{2}} + 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{5} c^{3} - 368 \, B a^{6} c^{\frac{5}{2}}}{120 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/x^7,x, algorithm="giac")

[Out]

5/8*A*c^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - B*c^(5/2)*l
n(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + 1/120*(165*(sqrt(c)*x - sqrt(c*x^2 + a))^
11*A*c^3 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^10*B*a*c^(5/2) + 25*(sqrt(c)*x - sq
rt(c*x^2 + a))^9*A*a*c^3 - 2160*(sqrt(c)*x - sqrt(c*x^2 + a))^8*B*a^2*c^(5/2) +
450*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*a^2*c^3 + 3680*(sqrt(c)*x - sqrt(c*x^2 + a
))^6*B*a^3*c^(5/2) + 450*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*a^3*c^3 - 3360*(sqrt(
c)*x - sqrt(c*x^2 + a))^4*B*a^4*c^(5/2) + 25*(sqrt(c)*x - sqrt(c*x^2 + a))^3*A*a
^4*c^3 + 1488*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a^5*c^(5/2) + 165*(sqrt(c)*x - s
qrt(c*x^2 + a))*A*a^5*c^3 - 368*B*a^6*c^(5/2))/((sqrt(c)*x - sqrt(c*x^2 + a))^2
- a)^6