3.381 \(\int \frac{A+B x}{x^3 \left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{5 A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A \sqrt{a+c x^2}}{2 a^3 x^2}-\frac{8 B \sqrt{a+c x^2}}{3 a^3 x}+\frac{5 A+4 B x}{3 a^2 x^2 \sqrt{a+c x^2}}+\frac{A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.35385, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{5 A c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A \sqrt{a+c x^2}}{2 a^3 x^2}-\frac{8 B \sqrt{a+c x^2}}{3 a^3 x}+\frac{5 A+4 B x}{3 a^2 x^2 \sqrt{a+c x^2}}+\frac{A+B x}{3 a x^2 \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^3*(a + c*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 46.5536, size = 119, normalized size = 0.92 \[ - \frac{5 A \sqrt{a + c x^{2}}}{2 a^{3} x^{2}} + \frac{5 A c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} - \frac{8 B \sqrt{a + c x^{2}}}{3 a^{3} x} + \frac{A + B x}{3 a x^{2} \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{5 A + 4 B x}{3 a^{2} x^{2} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**3/(c*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.369078, size = 103, normalized size = 0.8 \[ \frac{-\frac{\sqrt{a} \left (3 a^2 (A+2 B x)+4 a c x^2 (5 A+6 B x)+c^2 x^4 (15 A+16 B x)\right )}{x^2 \left (a+c x^2\right )^{3/2}}+15 A c \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-15 A c \log (x)}{6 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^3*(a + c*x^2)^(5/2)),x]

[Out]

(-((Sqrt[a]*(3*a^2*(A + 2*B*x) + 4*a*c*x^2*(5*A + 6*B*x) + c^2*x^4*(15*A + 16*B*
x)))/(x^2*(a + c*x^2)^(3/2))) - 15*A*c*Log[x] + 15*A*c*Log[a + Sqrt[a]*Sqrt[a +
c*x^2]])/(6*a^(7/2))

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Maple [A]  time = 0.014, size = 134, normalized size = 1. \[ -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ac}{6\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ac}{2\,{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{5\,Ac}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{B}{ax} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,Bcx}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,Bcx}{3\,{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^3/(c*x^2+a)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308087, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (16 \, B c^{2} x^{5} + 15 \, A c^{2} x^{4} + 24 \, B a c x^{3} + 20 \, A a c x^{2} + 6 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a} - 15 \,{\left (A c^{3} x^{6} + 2 \, A a c^{2} x^{4} + A a^{2} c x^{2}\right )} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{12 \,{\left (a^{3} c^{2} x^{6} + 2 \, a^{4} c x^{4} + a^{5} x^{2}\right )} \sqrt{a}}, -\frac{{\left (16 \, B c^{2} x^{5} + 15 \, A c^{2} x^{4} + 24 \, B a c x^{3} + 20 \, A a c x^{2} + 6 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a} - 15 \,{\left (A c^{3} x^{6} + 2 \, A a c^{2} x^{4} + A a^{2} c x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right )}{6 \,{\left (a^{3} c^{2} x^{6} + 2 \, a^{4} c x^{4} + a^{5} x^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[-1/12*(2*(16*B*c^2*x^5 + 15*A*c^2*x^4 + 24*B*a*c*x^3 + 20*A*a*c*x^2 + 6*B*a^2*x
+ 3*A*a^2)*sqrt(c*x^2 + a)*sqrt(a) - 15*(A*c^3*x^6 + 2*A*a*c^2*x^4 + A*a^2*c*x^
2)*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*x^2 + a)*a)/x^2))/((a^3*c^2*x^6 + 2*a^
4*c*x^4 + a^5*x^2)*sqrt(a)), -1/6*((16*B*c^2*x^5 + 15*A*c^2*x^4 + 24*B*a*c*x^3 +
20*A*a*c*x^2 + 6*B*a^2*x + 3*A*a^2)*sqrt(c*x^2 + a)*sqrt(-a) - 15*(A*c^3*x^6 +
2*A*a*c^2*x^4 + A*a^2*c*x^2)*arctan(sqrt(-a)/sqrt(c*x^2 + a)))/((a^3*c^2*x^6 + 2
*a^4*c*x^4 + a^5*x^2)*sqrt(-a))]

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Sympy [A]  time = 116.51, size = 1034, normalized size = 8.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**3/(c*x**2+a)**(5/2),x)

[Out]

A*(-6*a**17*sqrt(1 + c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**
(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) - 46*a**16*c*x**2*sqrt(1 + c*x**2/a)/
(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)
*c**3*x**8) - 15*a**16*c*x**2*log(c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*
x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) + 30*a**16*c*x**2*log(sq
rt(1 + c*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c*
*2*x**6 + 12*a**(33/2)*c**3*x**8) - 70*a**15*c**2*x**4*sqrt(1 + c*x**2/a)/(12*a*
*(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*
x**8) - 45*a**15*c**2*x**4*log(c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**
4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) + 90*a**15*c**2*x**4*log(sq
rt(1 + c*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c*
*2*x**6 + 12*a**(33/2)*c**3*x**8) - 30*a**14*c**3*x**6*sqrt(1 + c*x**2/a)/(12*a*
*(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*
x**8) - 45*a**14*c**3*x**6*log(c*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**
4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8) + 90*a**14*c**3*x**6*log(sq
rt(1 + c*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c*
*2*x**6 + 12*a**(33/2)*c**3*x**8) - 15*a**13*c**4*x**8*log(c*x**2/a)/(12*a**(39/
2)*x**2 + 36*a**(37/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8)
+ 30*a**13*c**4*x**8*log(sqrt(1 + c*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37
/2)*c*x**4 + 36*a**(35/2)*c**2*x**6 + 12*a**(33/2)*c**3*x**8)) + B*(-3*a**2*c**(
9/2)*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4 + 6*a**4*c**5*x**2 + 3*a**3*c**6*x**4) -
12*a*c**(11/2)*x**2*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4 + 6*a**4*c**5*x**2 + 3*a**
3*c**6*x**4) - 8*c**(13/2)*x**4*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4 + 6*a**4*c**5*
x**2 + 3*a**3*c**6*x**4))

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GIAC/XCAS [A]  time = 0.279503, size = 266, normalized size = 2.06 \[ -\frac{{\left ({\left (\frac{5 \, B c^{2} x}{a^{3}} + \frac{6 \, A c^{2}}{a^{3}}\right )} x + \frac{6 \, B c}{a^{2}}\right )} x + \frac{7 \, A c}{a^{2}}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} - \frac{5 \, A c \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a \sqrt{c} +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a c - 2 \, B a^{2} \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*x^3),x, algorithm="giac")

[Out]

-1/3*(((5*B*c^2*x/a^3 + 6*A*c^2/a^3)*x + 6*B*c/a^2)*x + 7*A*c/a^2)/(c*x^2 + a)^(
3/2) - 5*A*c*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^3) + ((
sqrt(c)*x - sqrt(c*x^2 + a))^3*A*c + 2*(sqrt(c)*x - sqrt(c*x^2 + a))^2*B*a*sqrt(
c) + (sqrt(c)*x - sqrt(c*x^2 + a))*A*a*c - 2*B*a^2*sqrt(c))/(((sqrt(c)*x - sqrt(
c*x^2 + a))^2 - a)^2*a^3)