3.325 \(\int \frac{(A+B x) \sqrt{a+c x^2}}{x^7} \, dx\)

Optimal. Leaf size=147 \[ -\frac{A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}-\frac{A c^2 \sqrt{a+c x^2}}{16 a^2 x^2}+\frac{A c \left (a+c x^2\right )^{3/2}}{8 a^2 x^4}+\frac{2 B c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac{A \left (a+c x^2\right )^{3/2}}{6 a x^6}-\frac{B \left (a+c x^2\right )^{3/2}}{5 a x^5} \]

[Out]

-(A*c^2*Sqrt[a + c*x^2])/(16*a^2*x^2) - (A*(a + c*x^2)^(3/2))/(6*a*x^6) - (B*(a
+ c*x^2)^(3/2))/(5*a*x^5) + (A*c*(a + c*x^2)^(3/2))/(8*a^2*x^4) + (2*B*c*(a + c*
x^2)^(3/2))/(15*a^2*x^3) - (A*c^3*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(16*a^(5/2))

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Rubi [A]  time = 0.352875, 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 \[ -\frac{A c^3 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}-\frac{A c^2 \sqrt{a+c x^2}}{16 a^2 x^2}+\frac{A c \left (a+c x^2\right )^{3/2}}{8 a^2 x^4}+\frac{2 B c \left (a+c x^2\right )^{3/2}}{15 a^2 x^3}-\frac{A \left (a+c x^2\right )^{3/2}}{6 a x^6}-\frac{B \left (a+c x^2\right )^{3/2}}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(A*c^2*Sqrt[a + c*x^2])/(16*a^2*x^2) - (A*(a + c*x^2)^(3/2))/(6*a*x^6) - (B*(a
+ c*x^2)^(3/2))/(5*a*x^5) + (A*c*(a + c*x^2)^(3/2))/(8*a^2*x^4) + (2*B*c*(a + c*
x^2)^(3/2))/(15*a^2*x^3) - (A*c^3*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(16*a^(5/2))

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Rubi in Sympy [A]  time = 42.1287, size = 134, normalized size = 0.91 \[ - \frac{A \left (a + c x^{2}\right )^{\frac{3}{2}}}{6 a x^{6}} - \frac{A c^{2} \sqrt{a + c x^{2}}}{16 a^{2} x^{2}} + \frac{A c \left (a + c x^{2}\right )^{\frac{3}{2}}}{8 a^{2} x^{4}} - \frac{A c^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}}} - \frac{B \left (a + c x^{2}\right )^{\frac{3}{2}}}{5 a x^{5}} + \frac{2 B c \left (a + c x^{2}\right )^{\frac{3}{2}}}{15 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(1/2)/x**7,x)

[Out]

-A*(a + c*x**2)**(3/2)/(6*a*x**6) - A*c**2*sqrt(a + c*x**2)/(16*a**2*x**2) + A*c
*(a + c*x**2)**(3/2)/(8*a**2*x**4) - A*c**3*atanh(sqrt(a + c*x**2)/sqrt(a))/(16*
a**(5/2)) - B*(a + c*x**2)**(3/2)/(5*a*x**5) + 2*B*c*(a + c*x**2)**(3/2)/(15*a**
2*x**3)

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Mathematica [A]  time = 0.194278, size = 114, normalized size = 0.78 \[ \frac{\sqrt{a} \sqrt{a+c x^2} \left (-8 a^2 (5 A+6 B x)-2 a c x^2 (5 A+8 B x)+c^2 x^4 (15 A+32 B x)\right )-15 A c^3 x^6 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )+15 A c^3 x^6 \log (x)}{240 a^{5/2} x^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[a + c*x^2]*(-8*a^2*(5*A + 6*B*x) - 2*a*c*x^2*(5*A + 8*B*x) + c^2*x
^4*(15*A + 32*B*x)) + 15*A*c^3*x^6*Log[x] - 15*A*c^3*x^6*Log[a + Sqrt[a]*Sqrt[a
+ c*x^2]])/(240*a^(5/2)*x^6)

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Maple [A]  time = 0.016, size = 147, normalized size = 1. \[ -{\frac{A}{6\,a{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ac}{8\,{a}^{2}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{A{c}^{2}}{16\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{A{c}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{A{c}^{3}}{16\,{a}^{3}}\sqrt{c{x}^{2}+a}}-{\frac{B}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{2\,Bc}{15\,{a}^{2}{x}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(1/2)/x^7,x)

[Out]

-1/6*A*(c*x^2+a)^(3/2)/a/x^6+1/8*A*c*(c*x^2+a)^(3/2)/a^2/x^4-1/16*A/a^3*c^2/x^2*
(c*x^2+a)^(3/2)-1/16*A/a^(5/2)*c^3*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)+1/16*A/
a^3*c^3*(c*x^2+a)^(1/2)-1/5*B*(c*x^2+a)^(3/2)/a/x^5+2/15*B*c*(c*x^2+a)^(3/2)/a^2
/x^3

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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(sqrt(c*x^2 + a)*(B*x + A)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.412329, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A c^{3} x^{6} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (32 \, B c^{2} x^{5} + 15 \, A c^{2} x^{4} - 16 \, B a c x^{3} - 10 \, A a c x^{2} - 48 \, B a^{2} x - 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{480 \, a^{\frac{5}{2}} x^{6}}, -\frac{15 \, A c^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (32 \, B c^{2} x^{5} + 15 \, A c^{2} x^{4} - 16 \, B a c x^{3} - 10 \, A a c x^{2} - 48 \, B a^{2} x - 40 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{240 \, \sqrt{-a} a^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/480*(15*A*c^3*x^6*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) + 2
*(32*B*c^2*x^5 + 15*A*c^2*x^4 - 16*B*a*c*x^3 - 10*A*a*c*x^2 - 48*B*a^2*x - 40*A*
a^2)*sqrt(c*x^2 + a)*sqrt(a))/(a^(5/2)*x^6), -1/240*(15*A*c^3*x^6*arctan(sqrt(-a
)/sqrt(c*x^2 + a)) - (32*B*c^2*x^5 + 15*A*c^2*x^4 - 16*B*a*c*x^3 - 10*A*a*c*x^2
- 48*B*a^2*x - 40*A*a^2)*sqrt(c*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^6)]

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Sympy [A]  time = 23.2628, size = 201, normalized size = 1.37 \[ - \frac{A a}{6 \sqrt{c} x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 A \sqrt{c}}{24 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{\frac{3}{2}}}{48 a x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A c^{\frac{5}{2}}}{16 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{A c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{16 a^{\frac{5}{2}}} - \frac{B \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a x^{2}} + \frac{2 B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(1/2)/x**7,x)

[Out]

-A*a/(6*sqrt(c)*x**7*sqrt(a/(c*x**2) + 1)) - 5*A*sqrt(c)/(24*x**5*sqrt(a/(c*x**2
) + 1)) + A*c**(3/2)/(48*a*x**3*sqrt(a/(c*x**2) + 1)) + A*c**(5/2)/(16*a**2*x*sq
rt(a/(c*x**2) + 1)) - A*c**3*asinh(sqrt(a)/(sqrt(c)*x))/(16*a**(5/2)) - B*sqrt(c
)*sqrt(a/(c*x**2) + 1)/(5*x**4) - B*c**(3/2)*sqrt(a/(c*x**2) + 1)/(15*a*x**2) +
2*B*c**(5/2)*sqrt(a/(c*x**2) + 1)/(15*a**2)

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GIAC/XCAS [A]  time = 0.27808, size = 439, normalized size = 2.99 \[ \frac{A c^{3} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{2}} - \frac{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} A c^{3} - 85 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} A a c^{3} - 480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} B a^{2} c^{\frac{5}{2}} - 570 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a^{2} c^{3} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{3} c^{\frac{5}{2}} - 570 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{3} c^{3} - 85 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{4} c^{3} + 192 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{5} c^{\frac{5}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{5} c^{3} - 32 \, B a^{6} c^{\frac{5}{2}}}{120 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{6} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*A*c^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 1/120
*(15*(sqrt(c)*x - sqrt(c*x^2 + a))^11*A*c^3 - 85*(sqrt(c)*x - sqrt(c*x^2 + a))^9
*A*a*c^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*B*a^2*c^(5/2) - 570*(sqrt(c)*x -
sqrt(c*x^2 + a))^7*A*a^2*c^3 + 320*(sqrt(c)*x - sqrt(c*x^2 + a))^6*B*a^3*c^(5/2)
- 570*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*a^3*c^3 - 85*(sqrt(c)*x - sqrt(c*x^2 +
a))^3*A*a^4*c^3 + 192*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a^5*c^(5/2) + 15*(sqrt(c
)*x - sqrt(c*x^2 + a))*A*a^5*c^3 - 32*B*a^6*c^(5/2))/(((sqrt(c)*x - sqrt(c*x^2 +
a))^2 - a)^6*a^2)