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

Optimal. Leaf size=317 \[ -\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 A}{6 a^2 x^{3/2}}-\frac{5 B}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a x^{3/2} \left (a+c x^2\right )} \]

[Out]

(-7*A)/(6*a^2*x^(3/2)) - (5*B)/(2*a^2*Sqrt[x]) + (A + B*x)/(2*a*x^(3/2)*(a + c*x
^2)) + ((5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])
/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) - ((5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(1/4)*ArcTan[1
+ (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) - ((5*Sqrt[a]*B - 7*A
*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8
*Sqrt[2]*a^(11/4)) + ((5*Sqrt[a]*B - 7*A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*
a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(11/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.717654, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 A}{6 a^2 x^{3/2}}-\frac{5 B}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a x^{3/2} \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-7*A)/(6*a^2*x^(3/2)) - (5*B)/(2*a^2*Sqrt[x]) + (A + B*x)/(2*a*x^(3/2)*(a + c*x
^2)) + ((5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])
/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) - ((5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(1/4)*ArcTan[1
+ (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) - ((5*Sqrt[a]*B - 7*A
*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8
*Sqrt[2]*a^(11/4)) + ((5*Sqrt[a]*B - 7*A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*
a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*a^(11/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 130.275, size = 301, normalized size = 0.95 \[ - \frac{7 A}{6 a^{2} x^{\frac{3}{2}}} - \frac{5 B}{2 a^{2} \sqrt{x}} + \frac{A + B x}{2 a x^{\frac{3}{2}} \left (a + c x^{2}\right )} + \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} - 5 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{11}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} - 5 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{11}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} + 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} + 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*A/(6*a**2*x**(3/2)) - 5*B/(2*a**2*sqrt(x)) + (A + B*x)/(2*a*x**(3/2)*(a + c*x
**2)) + sqrt(2)*c**(1/4)*(7*A*sqrt(c) - 5*B*sqrt(a))*log(-sqrt(2)*a**(1/4)*c**(3
/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(16*a**(11/4)) - sqrt(2)*c**(1/4)*(7*A*sqrt
(c) - 5*B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x
)/(16*a**(11/4)) + sqrt(2)*c**(1/4)*(7*A*sqrt(c) + 5*B*sqrt(a))*atan(1 - sqrt(2)
*c**(1/4)*sqrt(x)/a**(1/4))/(8*a**(11/4)) - sqrt(2)*c**(1/4)*(7*A*sqrt(c) + 5*B*
sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(8*a**(11/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.787859, size = 305, normalized size = 0.96 \[ \frac{3 \sqrt{2} \sqrt [4]{c} \left (7 \sqrt [4]{a} A \sqrt{c}-5 a^{3/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+3 \sqrt{2} \sqrt [4]{c} \left (5 a^{3/4} B-7 \sqrt [4]{a} A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-\frac{24 a c \sqrt{x} (A+B x)}{a+c x^2}+6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )-6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\frac{32 a A}{x^{3/2}}-\frac{96 a B}{\sqrt{x}}}{48 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a*A)/x^(3/2) - (96*a*B)/Sqrt[x] - (24*a*c*Sqrt[x]*(A + B*x))/(a + c*x^2) +
6*Sqrt[2]*a^(1/4)*(5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/
4)*Sqrt[x])/a^(1/4)] - 6*Sqrt[2]*a^(1/4)*(5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(1/4)*Arc
Tan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + 3*Sqrt[2]*(-5*a^(3/4)*B + 7*a^(1/4)
*A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] +
3*Sqrt[2]*(5*a^(3/4)*B - 7*a^(1/4)*A*Sqrt[c])*c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(
1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(48*a^3)

_______________________________________________________________________________________

Maple [A]  time = 0.026, size = 327, normalized size = 1. \[ -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{B}{{a}^{2}\sqrt{x}}}-{\frac{Bc}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}-{\frac{Ac}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }\sqrt{x}}-{\frac{7\,Ac\sqrt{2}}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{7\,Ac\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{7\,Ac\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{5\,B\sqrt{2}}{16\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,B\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,B\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.333849, size = 1235, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(60*B*c*x^3 + 28*A*c*x^2 + 48*B*a*x + 3*(a^2*c*x^3 + a^3*x)*sqrt(x)*sqrt(-
(a^5*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) + 70*A*B*c)
/a^5)*log(-(625*B^4*a^2*c - 2401*A^4*c^3)*sqrt(x) + (5*B*a^9*sqrt(-(625*B^4*a^2*
c - 2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) - 175*A*B^2*a^4*c + 343*A^3*a^3*c^2
)*sqrt(-(a^5*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) + 7
0*A*B*c)/a^5)) - 3*(a^2*c*x^3 + a^3*x)*sqrt(x)*sqrt(-(a^5*sqrt(-(625*B^4*a^2*c -
2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) + 70*A*B*c)/a^5)*log(-(625*B^4*a^2*c -
2401*A^4*c^3)*sqrt(x) - (5*B*a^9*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 + 24
01*A^4*c^3)/a^11) - 175*A*B^2*a^4*c + 343*A^3*a^3*c^2)*sqrt(-(a^5*sqrt(-(625*B^4
*a^2*c - 2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) + 70*A*B*c)/a^5)) - 3*(a^2*c*x
^3 + a^3*x)*sqrt(x)*sqrt((a^5*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 + 2401*A
^4*c^3)/a^11) - 70*A*B*c)/a^5)*log(-(625*B^4*a^2*c - 2401*A^4*c^3)*sqrt(x) + (5*
B*a^9*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) + 175*A*B^
2*a^4*c - 343*A^3*a^3*c^2)*sqrt((a^5*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 +
2401*A^4*c^3)/a^11) - 70*A*B*c)/a^5)) + 3*(a^2*c*x^3 + a^3*x)*sqrt(x)*sqrt((a^5
*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) - 70*A*B*c)/a^5
)*log(-(625*B^4*a^2*c - 2401*A^4*c^3)*sqrt(x) - (5*B*a^9*sqrt(-(625*B^4*a^2*c -
2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) + 175*A*B^2*a^4*c - 343*A^3*a^3*c^2)*sq
rt((a^5*sqrt(-(625*B^4*a^2*c - 2450*A^2*B^2*a*c^2 + 2401*A^4*c^3)/a^11) - 70*A*B
*c)/a^5)) + 16*A*a)/((a^2*c*x^3 + a^3*x)*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**(5/2)/(c*x**2+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284282, size = 393, normalized size = 1.24 \[ -\frac{B c x^{\frac{3}{2}} + A c \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} a^{2}} - \frac{2 \,{\left (3 \, B x + A\right )}}{3 \, a^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} c^{2}} - \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} c^{2}} - \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{3} c^{2}} + \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(B*c*x^(3/2) + A*c*sqrt(x))/((c*x^2 + a)*a^2) - 2/3*(3*B*x + A)/(a^2*x^(3/2
)) - 1/8*sqrt(2)*(7*(a*c^3)^(1/4)*A*c^2 + 5*(a*c^3)^(3/4)*B)*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)*(7*(a*c^3
)^(1/4)*A*c^2 + 5*(a*c^3)^(3/4)*B)*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)*(7*(a*c^3)^(1/4)*A*c^2 - 5*(a*c^3
)^(3/4)*B)*ln(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^3*c^2) + 1/16*sqrt
(2)*(7*(a*c^3)^(1/4)*A*c^2 - 5*(a*c^3)^(3/4)*B)*ln(-sqrt(2)*sqrt(x)*(a/c)^(1/4)
+ x + sqrt(a/c))/(a^3*c^2)