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

Optimal. Leaf size=320 \[ \frac{\left (5 \sqrt{a} B-21 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} B-21 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\sqrt{x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac{\sqrt{x} (A+B x)}{4 a \left (a+c x^2\right )^2} \]

[Out]

(Sqrt[x]*(A + B*x))/(4*a*(a + c*x^2)^2) + (Sqrt[x]*(7*A + 5*B*x))/(16*a^2*(a + c
*x^2)) - ((5*Sqrt[a]*B + 21*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1
/4)])/(32*Sqrt[2]*a^(11/4)*c^(3/4)) + ((5*Sqrt[a]*B + 21*A*Sqrt[c])*ArcTan[1 + (
Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(11/4)*c^(3/4)) + ((5*Sqrt[a]*B
- 21*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64
*Sqrt[2]*a^(11/4)*c^(3/4)) - ((5*Sqrt[a]*B - 21*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(11/4)*c^(3/4))

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Rubi [A]  time = 0.617652, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\left (5 \sqrt{a} B-21 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} B-21 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{11/4} c^{3/4}}-\frac{\left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{11/4} c^{3/4}}+\frac{\sqrt{x} (7 A+5 B x)}{16 a^2 \left (a+c x^2\right )}+\frac{\sqrt{x} (A+B x)}{4 a \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(A + B*x))/(4*a*(a + c*x^2)^2) + (Sqrt[x]*(7*A + 5*B*x))/(16*a^2*(a + c
*x^2)) - ((5*Sqrt[a]*B + 21*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1
/4)])/(32*Sqrt[2]*a^(11/4)*c^(3/4)) + ((5*Sqrt[a]*B + 21*A*Sqrt[c])*ArcTan[1 + (
Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(11/4)*c^(3/4)) + ((5*Sqrt[a]*B
- 21*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64
*Sqrt[2]*a^(11/4)*c^(3/4)) - ((5*Sqrt[a]*B - 21*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(11/4)*c^(3/4))

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Rubi in Sympy [A]  time = 118.339, size = 304, normalized size = 0.95 \[ \frac{\sqrt{x} \left (A + B x\right )}{4 a \left (a + c x^{2}\right )^{2}} + \frac{\sqrt{x} \left (\frac{7 A}{2} + \frac{5 B x}{2}\right )}{8 a^{2} \left (a + c x^{2}\right )} - \frac{\sqrt{2} \left (21 A \sqrt{c} - 5 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{11}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \left (21 A \sqrt{c} - 5 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{11}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \left (21 A \sqrt{c} + 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \left (21 A \sqrt{c} + 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*(A + B*x)/(4*a*(a + c*x**2)**2) + sqrt(x)*(7*A/2 + 5*B*x/2)/(8*a**2*(a +
c*x**2)) - sqrt(2)*(21*A*sqrt(c) - 5*B*sqrt(a))*log(-sqrt(2)*a**(1/4)*c**(3/4)*
sqrt(x) + sqrt(a)*sqrt(c) + c*x)/(128*a**(11/4)*c**(3/4)) + sqrt(2)*(21*A*sqrt(c
) - 5*B*sqrt(a))*log(sqrt(2)*a**(1/4)*c**(3/4)*sqrt(x) + sqrt(a)*sqrt(c) + c*x)/
(128*a**(11/4)*c**(3/4)) - sqrt(2)*(21*A*sqrt(c) + 5*B*sqrt(a))*atan(1 - sqrt(2)
*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(11/4)*c**(3/4)) + sqrt(2)*(21*A*sqrt(c) + 5*
B*sqrt(a))*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/a**(1/4))/(64*a**(11/4)*c**(3/4))

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Mathematica [A]  time = 0.639581, size = 311, normalized size = 0.97 \[ \frac{\frac{\sqrt{2} \left (5 a^{3/4} B-21 \sqrt [4]{a} A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{3/4}}+\frac{\sqrt{2} \left (21 \sqrt [4]{a} A \sqrt{c}-5 a^{3/4} B\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{3/4}}+\frac{32 a^2 \sqrt{x} (A+B x)}{\left (a+c x^2\right )^2}-\frac{2 \sqrt{2} \sqrt [4]{a} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{2 \sqrt{2} \sqrt [4]{a} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 a \sqrt{x} (7 A+5 B x)}{a+c x^2}}{128 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((32*a^2*Sqrt[x]*(A + B*x))/(a + c*x^2)^2 + (8*a*Sqrt[x]*(7*A + 5*B*x))/(a + c*x
^2) - (2*Sqrt[2]*a^(1/4)*(5*Sqrt[a]*B + 21*A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4
)*Sqrt[x])/a^(1/4)])/c^(3/4) + (2*Sqrt[2]*a^(1/4)*(5*Sqrt[a]*B + 21*A*Sqrt[c])*A
rcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(3/4) + (Sqrt[2]*(5*a^(3/4)*B -
21*a^(1/4)*A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]
)/c^(3/4) + (Sqrt[2]*(-5*a^(3/4)*B + 21*a^(1/4)*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(3/4))/(128*a^3)

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Maple [A]  time = 0.013, size = 349, normalized size = 1.1 \[{\frac{A}{4\,a \left ( c{x}^{2}+a \right ) ^{2}}\sqrt{x}}+{\frac{7\,A}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) }\sqrt{x}}+{\frac{21\,A\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{21\,A\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{21\,A\sqrt{2}}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{B}{4\,a \left ( c{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{5\,B}{16\,{a}^{2} \left ( c{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{5\,B\sqrt{2}}{128\,{a}^{2}c}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,B\sqrt{2}}{64\,{a}^{2}c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{5\,B\sqrt{2}}{64\,{a}^{2}c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*A*x^(1/2)/a/(c*x^2+a)^2+7/16*A/a^2*x^(1/2)/(c*x^2+a)+21/128*A/a^3*(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)))+21/64*A/a^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)
*x^(1/2)+1)+21/64*A/a^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1
)+1/4*B*x^(3/2)/a/(c*x^2+a)^2+5/16*B/a^2*x^(3/2)/(c*x^2+a)+5/128*B/a^2/c/(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*B/a^2/c/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(
1/4)*x^(1/2)+1)+5/64*B/a^2/c/(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)/((c*x^2 + a)^3*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319169, size = 1324, normalized size = 4.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*((a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)*sqrt(-(a^5*c*sqrt(-(625*B^4*a^2 - 22050*
A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 210*A*B)/(a^5*c))*log(-(625*B^4*a^2
- 194481*A^4*c^2)*sqrt(x) + (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c
+ 194481*A^4*c^2)/(a^11*c^3)) - 525*A*B^2*a^4*c + 9261*A^3*a^3*c^2)*sqrt(-(a^5*c
*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 210*A*B)
/(a^5*c))) - (a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)*sqrt(-(a^5*c*sqrt(-(625*B^4*a^2 -
22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 210*A*B)/(a^5*c))*log(-(625*B
^4*a^2 - 194481*A^4*c^2)*sqrt(x) - (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 - 22050*A^2*B
^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) - 525*A*B^2*a^4*c + 9261*A^3*a^3*c^2)*sqrt(
-(a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 2
10*A*B)/(a^5*c))) - (a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)*sqrt((a^5*c*sqrt(-(625*B^4
*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) - 210*A*B)/(a^5*c))*log(-
(625*B^4*a^2 - 194481*A^4*c^2)*sqrt(x) + (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 - 22050
*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 525*A*B^2*a^4*c - 9261*A^3*a^3*c^2)
*sqrt((a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)
) - 210*A*B)/(a^5*c))) + (a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)*sqrt((a^5*c*sqrt(-(62
5*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) - 210*A*B)/(a^5*c))*
log(-(625*B^4*a^2 - 194481*A^4*c^2)*sqrt(x) - (5*B*a^9*c^2*sqrt(-(625*B^4*a^2 -
22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11*c^3)) + 525*A*B^2*a^4*c - 9261*A^3*a^3
*c^2)*sqrt((a^5*c*sqrt(-(625*B^4*a^2 - 22050*A^2*B^2*a*c + 194481*A^4*c^2)/(a^11
*c^3)) - 210*A*B)/(a^5*c))) + 4*(5*B*c*x^3 + 7*A*c*x^2 + 9*B*a*x + 11*A*a)*sqrt(
x))/(a^2*c^2*x^4 + 2*a^3*c*x^2 + a^4)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296122, size = 396, normalized size = 1.24 \[ \frac{5 \, B c x^{\frac{7}{2}} + 7 \, A c x^{\frac{5}{2}} + 9 \, B a x^{\frac{3}{2}} + 11 \, A a \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} a^{2}} + \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} c^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{3} c^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(5*B*c*x^(7/2) + 7*A*c*x^(5/2) + 9*B*a*x^(3/2) + 11*A*a*sqrt(x))/((c*x^2 +
a)^2*a^2) + 1/64*sqrt(2)*(21*(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^3) + 1/64*sqrt(2)
*(21*(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^3) + 1/128*sqrt(2)*(21*(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^3
) - 1/128*sqrt(2)*(21*(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^3)