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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.614333, 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 \[ \frac{\left (5 \sqrt{a} B+3 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{a} c^{9/4}}-\frac{\left (5 \sqrt{a} B+3 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} \sqrt [4]{a} c^{9/4}}+\frac{\left (5 \sqrt{a} B-3 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} c^{9/4}}-\frac{\left (5 \sqrt{a} B-3 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} c^{9/4}}-\frac{x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac{5 B \sqrt{x}}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 107.867, size = 291, normalized size = 0.96 \[ \frac{5 B \sqrt{x}}{2 c^{2}} - \frac{x^{\frac{3}{2}} \left (2 A + 2 B x\right )}{4 c \left (a + c x^{2}\right )} - \frac{\sqrt{2} \left (3 A \sqrt{c} - 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} c^{\frac{9}{4}}} + \frac{\sqrt{2} \left (3 A \sqrt{c} - 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} c^{\frac{9}{4}}} + \frac{\sqrt{2} \left (3 A \sqrt{c} + 5 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 \sqrt [4]{a} c^{\frac{9}{4}}} - \frac{\sqrt{2} \left (3 A \sqrt{c} + 5 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 \sqrt [4]{a} c^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.781735, size = 292, normalized size = 0.96 \[ \frac{-\frac{8 c^{3/4} \sqrt{x} (A c x-a B)}{a+c x^2}+\frac{\sqrt{2} \left (5 \sqrt{a} B \sqrt{c}+3 A c\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{a}}-\frac{\sqrt{2} \left (5 \sqrt{a} B \sqrt{c}+3 A c\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{\sqrt [4]{a}}+\frac{2 \sqrt{2} \left (5 \sqrt{a} B \sqrt{c}-3 A c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}-\frac{2 \sqrt{2} \left (5 \sqrt{a} B \sqrt{c}-3 A c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}+32 B c^{3/4} \sqrt{x}}{16 c^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

(32*B*c^(3/4)*Sqrt[x] - (8*c^(3/4)*Sqrt[x]*(-(a*B) + A*c*x))/(a + c*x^2) + (2*Sq
rt[2]*(5*Sqrt[a]*B*Sqrt[c] - 3*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)
])/a^(1/4) - (2*Sqrt[2]*(5*Sqrt[a]*B*Sqrt[c] - 3*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4
)*Sqrt[x])/a^(1/4)])/a^(1/4) + (Sqrt[2]*(5*Sqrt[a]*B*Sqrt[c] + 3*A*c)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/a^(1/4) - (Sqrt[2]*(5*Sqrt[a]*
B*Sqrt[c] + 3*A*c)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/a
^(1/4))/(16*c^(11/4))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 314, normalized size = 1. \[ 2\,{\frac{B\sqrt{x}}{{c}^{2}}}-{\frac{A}{2\,c \left ( c{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{Ba}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }\sqrt{x}}-{\frac{5\,B\sqrt{2}}{8\,{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{5\,B\sqrt{2}}{16\,{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{5\,B\sqrt{2}}{8\,{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,A\sqrt{2}}{16\,{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,A\sqrt{2}}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,A\sqrt{2}}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.344726, size = 1193, normalized size = 3.92 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((c^3*x^2 + a*c^2)*sqrt((c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c
^2)/(a*c^9)) + 30*A*B)/c^4)*log(-(625*B^4*a^2 - 81*A^4*c^2)*sqrt(x) + (3*A*a*c^7
*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) + 125*B^3*a^2*c^2 -
45*A^2*B*a*c^3)*sqrt((c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a
*c^9)) + 30*A*B)/c^4)) - (c^3*x^2 + a*c^2)*sqrt((c^4*sqrt(-(625*B^4*a^2 - 450*A^
2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) + 30*A*B)/c^4)*log(-(625*B^4*a^2 - 81*A^4*c^2)*
sqrt(x) - (3*A*a*c^7*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9))
+ 125*B^3*a^2*c^2 - 45*A^2*B*a*c^3)*sqrt((c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*
a*c + 81*A^4*c^2)/(a*c^9)) + 30*A*B)/c^4)) - (c^3*x^2 + a*c^2)*sqrt(-(c^4*sqrt(-
(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 30*A*B)/c^4)*log(-(625*B
^4*a^2 - 81*A^4*c^2)*sqrt(x) + (3*A*a*c^7*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c +
81*A^4*c^2)/(a*c^9)) - 125*B^3*a^2*c^2 + 45*A^2*B*a*c^3)*sqrt(-(c^4*sqrt(-(625*
B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 30*A*B)/c^4)) + (c^3*x^2 + a*
c^2)*sqrt(-(c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 30
*A*B)/c^4)*log(-(625*B^4*a^2 - 81*A^4*c^2)*sqrt(x) - (3*A*a*c^7*sqrt(-(625*B^4*a
^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 125*B^3*a^2*c^2 + 45*A^2*B*a*c^3)*
sqrt(-(c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 30*A*B)
/c^4)) + 4*(4*B*c*x^2 - A*c*x + 5*B*a)*sqrt(x))/(c^3*x^2 + a*c^2)

_______________________________________________________________________________________

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(x**(5/2)*(B*x+A)/(c*x**2+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.283401, size = 396, normalized size = 1.3 \[ \frac{2 \, B \sqrt{x}}{c^{2}} - \frac{A c x^{\frac{3}{2}} - B a \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} c^{2}} - \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a c^{4}} + \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a c^{4}} - \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a c^{6}} - \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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