3.419 \(\int \frac{A+B x}{x^{9/2} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=306 \[ \frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{11/4}}+\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{11/4}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.782359, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{11/4}}-\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{11/4}}+\frac{c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{11/4}}+\frac{2 A c}{3 a^2 x^{3/2}}+\frac{2 B c}{a^2 \sqrt{x}}-\frac{2 A}{7 a x^{7/2}}-\frac{2 B}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(9/2)*(a + c*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 131.093, size = 292, normalized size = 0.95 \[ - \frac{2 A}{7 a x^{\frac{7}{2}}} + \frac{2 A c}{3 a^{2} x^{\frac{3}{2}}} - \frac{2 B}{5 a x^{\frac{5}{2}}} + \frac{2 B c}{a^{2} \sqrt{x}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{11}{4}}} - \frac{\sqrt{2} c^{\frac{5}{4}} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(9/2)/(c*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.811248, size = 300, normalized size = 0.98 \[ \frac{105 \sqrt{2} c^{5/4} \left (a^{3/4} B-\sqrt [4]{a} A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+105 \sqrt{2} c^{5/4} \left (\sqrt [4]{a} A \sqrt{c}-a^{3/4} B\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-\frac{120 a^2 A}{x^{7/2}}-\frac{168 a^2 B}{x^{5/2}}-210 \sqrt{2} \sqrt [4]{a} c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )+210 \sqrt{2} \sqrt [4]{a} c^{5/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{280 a A c}{x^{3/2}}+\frac{840 a B c}{\sqrt{x}}}{420 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(9/2)*(a + c*x^2)),x]

[Out]

((-120*a^2*A)/x^(7/2) - (168*a^2*B)/x^(5/2) + (280*a*A*c)/x^(3/2) + (840*a*B*c)/
Sqrt[x] - 210*Sqrt[2]*a^(1/4)*(Sqrt[a]*B + A*Sqrt[c])*c^(5/4)*ArcTan[1 - (Sqrt[2
]*c^(1/4)*Sqrt[x])/a^(1/4)] + 210*Sqrt[2]*a^(1/4)*(Sqrt[a]*B + A*Sqrt[c])*c^(5/4
)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)] + 105*Sqrt[2]*(a^(3/4)*B - a^(1/
4)*A*Sqrt[c])*c^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]
+ 105*Sqrt[2]*(-(a^(3/4)*B) + a^(1/4)*A*Sqrt[c])*c^(5/4)*Log[Sqrt[a] + Sqrt[2]*
a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(420*a^3)

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Maple [A]  time = 0.016, size = 318, normalized size = 1. \[ -{\frac{2\,A}{7\,a}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{5\,a}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Ac}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Bc}{{a}^{2}\sqrt{x}}}+{\frac{A{c}^{2}\sqrt{2}}{4\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{A{c}^{2}\sqrt{2}}{2\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{A{c}^{2}\sqrt{2}}{2\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{Bc\sqrt{2}}{4\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{Bc\sqrt{2}}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{Bc\sqrt{2}}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(9/2)/(c*x^2+a),x)

[Out]

-2/7*A/a/x^(7/2)-2/5*B/a/x^(5/2)+2/3*A*c/a^2/x^(3/2)+2*B*c/a^2/x^(1/2)+1/4/a^3*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)))+1/2/a^3*c^2*A*(a/c)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(a/c)^(1/4)*x^(1/2)+1)+1/2/a^3*c^2*A*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a
/c)^(1/4)*x^(1/2)-1)+1/4/a^2*c*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)))+1/2/a^2*c*B/(a/
c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+1/2/a^2*c*B/(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)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.328341, size = 1185, normalized size = 3.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*x^(9/2)),x, algorithm="fricas")

[Out]

1/210*(105*a^2*x^(7/2)*sqrt(-(a^5*sqrt(-(B^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7
)/a^11) + 2*A*B*c^3)/a^5)*log(-(B^4*a^2*c^4 - A^4*c^6)*sqrt(x) + (B*a^9*sqrt(-(B
^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) - A*B^2*a^4*c^3 + A^3*a^3*c^4)*sqr
t(-(a^5*sqrt(-(B^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) + 2*A*B*c^3)/a^5))
- 105*a^2*x^(7/2)*sqrt(-(a^5*sqrt(-(B^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^
11) + 2*A*B*c^3)/a^5)*log(-(B^4*a^2*c^4 - A^4*c^6)*sqrt(x) - (B*a^9*sqrt(-(B^4*a
^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) - A*B^2*a^4*c^3 + A^3*a^3*c^4)*sqrt(-(
a^5*sqrt(-(B^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) + 2*A*B*c^3)/a^5)) - 1
05*a^2*x^(7/2)*sqrt((a^5*sqrt(-(B^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) -
2*A*B*c^3)/a^5)*log(-(B^4*a^2*c^4 - A^4*c^6)*sqrt(x) + (B*a^9*sqrt(-(B^4*a^2*c^
5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) + A*B^2*a^4*c^3 - A^3*a^3*c^4)*sqrt((a^5*sq
rt(-(B^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) - 2*A*B*c^3)/a^5)) + 105*a^2
*x^(7/2)*sqrt((a^5*sqrt(-(B^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) - 2*A*B
*c^3)/a^5)*log(-(B^4*a^2*c^4 - A^4*c^6)*sqrt(x) - (B*a^9*sqrt(-(B^4*a^2*c^5 - 2*
A^2*B^2*a*c^6 + A^4*c^7)/a^11) + A*B^2*a^4*c^3 - A^3*a^3*c^4)*sqrt((a^5*sqrt(-(B
^4*a^2*c^5 - 2*A^2*B^2*a*c^6 + A^4*c^7)/a^11) - 2*A*B*c^3)/a^5)) + 420*B*c*x^3 +
140*A*c*x^2 - 84*B*a*x - 60*A*a)/(a^2*x^(7/2))

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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**(9/2)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284243, size = 373, normalized size = 1.22 \[ \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{3} c} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{3} c} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{3} c} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{3} c} + \frac{2 \,{\left (105 \, B c x^{3} + 35 \, A c x^{2} - 21 \, B a x - 15 \, A a\right )}}{105 \, a^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*x^(9/2)),x, algorithm="giac")

[Out]

1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + (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) + 1/2*sqrt(2)*((a*c^3)^(1/4)*A*c^2
+ (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) + 1/4*sqrt(2)*((a*c^3)^(1/4)*A*c^2 - (a*c^3)^(3/4)*B)*ln(sqrt(2)*
sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^3*c) - 1/4*sqrt(2)*((a*c^3)^(1/4)*A*c^2
- (a*c^3)^(3/4)*B)*ln(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^3*c) + 2/
105*(105*B*c*x^3 + 35*A*c*x^2 - 21*B*a*x - 15*A*a)/(a^2*x^(7/2))