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

Optimal. Leaf size=304 \[ -\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )} \]

[Out]

(-5*A)/(2*a^2*Sqrt[x]) + (A + B*x)/(2*a*Sqrt[x]*(a + c*x^2)) - ((3*Sqrt[a]*B - 5
*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*c^
(1/4)) + ((3*Sqrt[a]*B - 5*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/
4)])/(4*Sqrt[2]*a^(9/4)*c^(1/4)) - ((3*Sqrt[a]*B + 5*A*Sqrt[c])*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(9/4)*c^(1/4)) + ((3*Sq
rt[a]*B + 5*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/(8*Sqrt[2]*a^(9/4)*c^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.610836, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}+\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} \sqrt [4]{c}}-\frac{5 A}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a \sqrt{x} \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-5*A)/(2*a^2*Sqrt[x]) + (A + B*x)/(2*a*Sqrt[x]*(a + c*x^2)) - ((3*Sqrt[a]*B - 5
*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*c^
(1/4)) + ((3*Sqrt[a]*B - 5*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/
4)])/(4*Sqrt[2]*a^(9/4)*c^(1/4)) - ((3*Sqrt[a]*B + 5*A*Sqrt[c])*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(9/4)*c^(1/4)) + ((3*Sq
rt[a]*B + 5*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/(8*Sqrt[2]*a^(9/4)*c^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 116.541, size = 287, normalized size = 0.94 \[ - \frac{5 A}{2 a^{2} \sqrt{x}} + \frac{A + B x}{2 a \sqrt{x} \left (a + c x^{2}\right )} + \frac{\sqrt{2} \left (5 A \sqrt{c} - 3 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (5 A \sqrt{c} - 3 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} \left (5 A \sqrt{c} + 3 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{9}{4}} \sqrt [4]{c}} + \frac{\sqrt{2} \left (5 A \sqrt{c} + 3 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{9}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*A/(2*a**2*sqrt(x)) + (A + B*x)/(2*a*sqrt(x)*(a + c*x**2)) + sqrt(2)*(5*A*sqrt
(c) - 3*B*sqrt(a))*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(8*a**(9/4)*c**(1
/4)) - sqrt(2)*(5*A*sqrt(c) - 3*B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**
(1/4))/(8*a**(9/4)*c**(1/4)) - sqrt(2)*(5*A*sqrt(c) + 3*B*sqrt(a))*log(-sqrt(2)*
a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(16*a**(9/4)*c**(1/4)) + sqrt
(2)*(5*A*sqrt(c) + 3*B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*
sqrt(c) + c*x)/(16*a**(9/4)*c**(1/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.816715, size = 298, normalized size = 0.98 \[ \frac{-\frac{\sqrt{2} \left (5 a^{3/4} A \sqrt{c}+3 a^{5/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{\sqrt{2} \left (5 a^{3/4} A \sqrt{c}+3 a^{5/4} B\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{2 \sqrt{2} a^{3/4} \left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac{2 \sqrt{2} a^{3/4} \left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{c}}+\frac{8 a \sqrt{x} (a B-A c x)}{a+c x^2}-\frac{32 a A}{\sqrt{x}}}{16 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a*A)/Sqrt[x] + (8*a*Sqrt[x]*(a*B - A*c*x))/(a + c*x^2) - (2*Sqrt[2]*a^(3/4
)*(3*Sqrt[a]*B - 5*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(
1/4) + (2*Sqrt[2]*a^(3/4)*(3*Sqrt[a]*B - 5*A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4
)*Sqrt[x])/a^(1/4)])/c^(1/4) - (Sqrt[2]*(3*a^(5/4)*B + 5*a^(3/4)*A*Sqrt[c])*Log[
Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(1/4) + (Sqrt[2]*(3*a^
(5/4)*B + 5*a^(3/4)*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + S
qrt[c]*x])/c^(1/4))/(16*a^3)

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 314, normalized size = 1. \[ -2\,{\frac{A}{{a}^{2}\sqrt{x}}}-{\frac{Ac}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{B}{2\,a \left ( c{x}^{2}+a \right ) }\sqrt{x}}+{\frac{3\,B\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,B\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{3\,B\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{5\,A\sqrt{2}}{16\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,A\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,A\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.318614, size = 1183, normalized size = 3.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(20*A*c*x^2 - 4*B*a*x + (a^2*c*x^2 + a^3)*sqrt(x)*sqrt((a^4*sqrt(-(81*B^4*a
^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 30*A*B)/a^4)*log(-(81*B^4*a^2 - 6
25*A^4*c^2)*sqrt(x) + (5*A*a^7*c*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c
^2)/(a^9*c)) + 27*B^3*a^4 - 75*A^2*B*a^3*c)*sqrt((a^4*sqrt(-(81*B^4*a^2 - 450*A^
2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 30*A*B)/a^4)) - (a^2*c*x^2 + a^3)*sqrt(x)*sq
rt((a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 30*A*B)/a^
4)*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (5*A*a^7*c*sqrt(-(81*B^4*a^2 - 450*
A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 27*B^3*a^4 - 75*A^2*B*a^3*c)*sqrt((a^4*sqr
t(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) + 30*A*B)/a^4)) - (a^2*
c*x^2 + a^3)*sqrt(x)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^
2)/(a^9*c)) - 30*A*B)/a^4)*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) + (5*A*a^7*c*
sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) - 27*B^3*a^4 + 75*A^
2*B*a^3*c)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)
) - 30*A*B)/a^4)) + (a^2*c*x^2 + a^3)*sqrt(x)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450
*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c)) - 30*A*B)/a^4)*log(-(81*B^4*a^2 - 625*A^4*c
^2)*sqrt(x) - (5*A*a^7*c*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9
*c)) - 27*B^3*a^4 + 75*A^2*B*a^3*c)*sqrt(-(a^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a
*c + 625*A^4*c^2)/(a^9*c)) - 30*A*B)/a^4)) + 16*A*a)/((a^2*c*x^2 + a^3)*sqrt(x))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.288038, size = 379, normalized size = 1.25 \[ -\frac{5 \, A c x^{2} - B a x + 4 \, A a}{2 \,{\left (c x^{\frac{5}{2}} + a \sqrt{x}\right )} a^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} c^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} c^{2}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{3} c^{2}} - \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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