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

Optimal. Leaf size=331 \[ -\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{9/4} c^{5/4}}+\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{9/4} c^{5/4}}-\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{9/4} c^{5/4}}+\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{9/4} c^{5/4}}+\frac{\sqrt{x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac{\sqrt{x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-(Sqrt[x]*(a*B - A*c*x))/(4*a*c*(a + c*x^2)^2) + (Sqrt[x]*(a*B + 5*A*c*x))/(16*a
^2*c*(a + c*x^2)) - ((3*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqr
t[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*c^(5/4)) + ((3*Sqrt[a]*B + 5*A*Sqrt[c])*ArcT
an[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*c^(5/4)) - ((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
])/(64*Sqrt[2]*a^(9/4)*c^(5/4)) + ((3*Sqrt[a]*B - 5*A*Sqrt[c])*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(9/4)*c^(5/4))

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Rubi [A]  time = 0.607838, antiderivative size = 331, 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 \[ -\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{9/4} c^{5/4}}+\frac{\left (3 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{9/4} c^{5/4}}-\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{9/4} c^{5/4}}+\frac{\left (3 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{9/4} c^{5/4}}+\frac{\sqrt{x} (a B+5 A c x)}{16 a^2 c \left (a+c x^2\right )}-\frac{\sqrt{x} (a B-A c x)}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x]*(a*B - A*c*x))/(4*a*c*(a + c*x^2)^2) + (Sqrt[x]*(a*B + 5*A*c*x))/(16*a
^2*c*(a + c*x^2)) - ((3*Sqrt[a]*B + 5*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqr
t[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*c^(5/4)) + ((3*Sqrt[a]*B + 5*A*Sqrt[c])*ArcT
an[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(9/4)*c^(5/4)) - ((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
])/(64*Sqrt[2]*a^(9/4)*c^(5/4)) + ((3*Sqrt[a]*B - 5*A*Sqrt[c])*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(9/4)*c^(5/4))

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Rubi in Sympy [A]  time = 116.718, size = 313, normalized size = 0.95 \[ - \frac{\sqrt{x} \left (- A c x + B a\right )}{4 a c \left (a + c x^{2}\right )^{2}} + \frac{\sqrt{x} \left (\frac{5 A c x}{2} + \frac{B a}{2}\right )}{8 a^{2} c \left (a + c x^{2}\right )} + \frac{\sqrt{2} \left (5 A \sqrt{c} - 3 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{9}{4}} c^{\frac{5}{4}}} - \frac{\sqrt{2} \left (5 A \sqrt{c} - 3 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{9}{4}} c^{\frac{5}{4}}} - \frac{\sqrt{2} \left (5 A \sqrt{c} + 3 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}} c^{\frac{5}{4}}} + \frac{\sqrt{2} \left (5 A \sqrt{c} + 3 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)*(B*x+A)/(c*x**2+a)**3,x)

[Out]

-sqrt(x)*(-A*c*x + B*a)/(4*a*c*(a + c*x**2)**2) + sqrt(x)*(5*A*c*x/2 + B*a/2)/(8
*a**2*c*(a + c*x**2)) + 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)/(128*a**(9/4)*c**(5/4)) - 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)/(128*a**(9/4)*c**(5/4)) - sqrt(2)*(5*A*sqrt(c) + 3*B*sqrt(a))*atan(1 -
sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(9/4)*c**(5/4)) + sqrt(2)*(5*A*sqrt(c)
+ 3*B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(9/4)*c**(5/4
))

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Mathematica [A]  time = 0.596941, size = 315, normalized size = 0.95 \[ \frac{\sqrt{2} \left (5 a^{3/4} A c-3 a^{5/4} B \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+\sqrt{2} \left (3 a^{5/4} B \sqrt{c}-5 a^{3/4} A c\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-2 \sqrt{2} \left (5 a^{3/4} A c+3 a^{5/4} B \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} \left (5 a^{3/4} A c+3 a^{5/4} B \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{32 a^2 c^{3/4} \sqrt{x} (A c x-a B)}{\left (a+c x^2\right )^2}+\frac{8 a c^{3/4} \sqrt{x} (a B+5 A c x)}{a+c x^2}}{128 a^3 c^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((32*a^2*c^(3/4)*Sqrt[x]*(-(a*B) + A*c*x))/(a + c*x^2)^2 + (8*a*c^(3/4)*Sqrt[x]*
(a*B + 5*A*c*x))/(a + c*x^2) - 2*Sqrt[2]*(3*a^(5/4)*B*Sqrt[c] + 5*a^(3/4)*A*c)*A
rcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*(3*a^(5/4)*B*Sqrt[c] +
5*a^(3/4)*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + Sqrt[2]*(-3*a^(5/
4)*B*Sqrt[c] + 5*a^(3/4)*A*c)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sq
rt[c]*x] + Sqrt[2]*(3*a^(5/4)*B*Sqrt[c] - 5*a^(3/4)*A*c)*Log[Sqrt[a] + Sqrt[2]*a
^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(128*a^3*c^(7/4))

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Maple [A]  time = 0.022, size = 335, normalized size = 1. \[ 2\,{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{5\,Ac{x}^{7/2}}{32\,{a}^{2}}}+1/32\,{\frac{B{x}^{5/2}}{a}}+{\frac{9\,A{x}^{3/2}}{32\,a}}-{\frac{3\,B\sqrt{x}}{32\,c}} \right ) }+{\frac{3\,B\sqrt{2}}{128\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,B\sqrt{2}}{64\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,B\sqrt{2}}{64\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{5\,A\sqrt{2}}{128\,{a}^{2}c}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,A\sqrt{2}}{64\,{a}^{2}c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,A\sqrt{2}}{64\,{a}^{2}c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(B*x+A)/(c*x^2+a)^3,x)

[Out]

2*(5/32*A*c/a^2*x^(7/2)+1/32*B/a*x^(5/2)+9/32*A/a*x^(3/2)-3/32*B*x^(1/2)/c)/(c*x
^2+a)^2+3/128*B/a^2*(a/c)^(1/4)/c*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)))+3/64*B/a^2*(a/c)^(1/4)/c*2
^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+3/64*B/a^2*(a/c)^(1/4)/c*2^(1/2)*ar
ctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+5/128*A/c/a^2/(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))
)+5/64*A/c/a^2/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+5/64*A/
c/a^2/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)

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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)*sqrt(x)/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306919, size = 1380, normalized size = 4.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*((a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^4*c^2*sqrt(-(81*B^4*a^2 -
450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) + 30*A*B)/(a^4*c^2))*log(-(81*B^4*a^2
- 625*A^4*c^2)*sqrt(x) + (5*A*a^7*c^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*
A^4*c^2)/(a^9*c^5)) + 27*B^3*a^4*c - 75*A^2*B*a^3*c^2)*sqrt(-(a^4*c^2*sqrt(-(81*
B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) + 30*A*B)/(a^4*c^2))) - (a^2
*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2
*a*c + 625*A^4*c^2)/(a^9*c^5)) + 30*A*B)/(a^4*c^2))*log(-(81*B^4*a^2 - 625*A^4*c
^2)*sqrt(x) - (5*A*a^7*c^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a
^9*c^5)) + 27*B^3*a^4*c - 75*A^2*B*a^3*c^2)*sqrt(-(a^4*c^2*sqrt(-(81*B^4*a^2 - 4
50*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) + 30*A*B)/(a^4*c^2))) - (a^2*c^3*x^4 +
2*a^3*c^2*x^2 + a^4*c)*sqrt((a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A
^4*c^2)/(a^9*c^5)) - 30*A*B)/(a^4*c^2))*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x)
+ (5*A*a^7*c^4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) - 2
7*B^3*a^4*c + 75*A^2*B*a^3*c^2)*sqrt((a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*
c + 625*A^4*c^2)/(a^9*c^5)) - 30*A*B)/(a^4*c^2))) + (a^2*c^3*x^4 + 2*a^3*c^2*x^2
+ a^4*c)*sqrt((a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*
c^5)) - 30*A*B)/(a^4*c^2))*log(-(81*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (5*A*a^7*c^
4*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c^2)/(a^9*c^5)) - 27*B^3*a^4*c +
75*A^2*B*a^3*c^2)*sqrt((a^4*c^2*sqrt(-(81*B^4*a^2 - 450*A^2*B^2*a*c + 625*A^4*c
^2)/(a^9*c^5)) - 30*A*B)/(a^4*c^2))) - 4*(5*A*c^2*x^3 + B*a*c*x^2 + 9*A*a*c*x -
3*B*a^2)*sqrt(x))/(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)

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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**(1/2)*(B*x+A)/(c*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29725, size = 414, normalized size = 1.25 \[ \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} c^{3}} - \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{3} c^{3}} + \frac{5 \, A c^{2} x^{\frac{7}{2}} + B a c x^{\frac{5}{2}} + 9 \, A a c x^{\frac{3}{2}} - 3 \, B a^{2} \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} c^{5}} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} B a c^{3} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} A c^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{3} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c + 5*(a*c^3)^(3/4)*A)*arctan(-1/2*sqrt(2)*(sq
rt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^3*c^3) - 1/128*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^3) + 1/16*(5*A*c^2*x^(7/2) + B*a*c*x^(5/2) + 9*A*a*c*x^(3/2) - 3*B*a^2
*sqrt(x))/((c*x^2 + a)^2*a^2*c) + 1/64*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c^3 + 5*(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^3*c^5) + 1/128*sqrt(2)*(3*(a*c^3)^(1/4)*B*a*c^3 - 5*(a*c^3)^(3/4)*A*c^2)*l
n(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^3*c^5)