3.418 \(\int \frac{A+B x}{x^{7/2} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=292 \[ \frac{c^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{c^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{9/4}}+\frac{c^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}-\frac{c^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4}}+\frac{2 A c}{a^2 \sqrt{x}}-\frac{2 A}{5 a x^{5/2}}-\frac{2 B}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.696486, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{c^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{c^{3/4} \left (\sqrt{a} B+A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{9/4}}+\frac{c^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}-\frac{c^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4}}+\frac{2 A c}{a^2 \sqrt{x}}-\frac{2 A}{5 a x^{5/2}}-\frac{2 B}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*(a + c*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 119.143, size = 277, normalized size = 0.95 \[ - \frac{2 A}{5 a x^{\frac{5}{2}}} + \frac{2 A c}{a^{2} \sqrt{x}} - \frac{2 B}{3 a x^{\frac{3}{2}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{9}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (A \sqrt{c} - B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{9}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{9}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (A \sqrt{c} + B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(7/2)/(c*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.940745, size = 290, normalized size = 0.99 \[ \frac{15 \sqrt{2} c^{3/4} \left (a^{3/4} A \sqrt{c}+a^{5/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-15 \sqrt{2} c^{3/4} \left (a^{3/4} A \sqrt{c}+a^{5/4} B\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+30 \sqrt{2} a^{3/4} c^{3/4} \left (\sqrt{a} B-A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )+30 \sqrt{2} a^{3/4} c^{3/4} \left (A \sqrt{c}-\sqrt{a} B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\frac{24 a^2 A}{x^{5/2}}-\frac{40 a^2 B}{x^{3/2}}+\frac{120 a A c}{\sqrt{x}}}{60 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*(a + c*x^2)),x]

[Out]

((-24*a^2*A)/x^(5/2) - (40*a^2*B)/x^(3/2) + (120*a*A*c)/Sqrt[x] + 30*Sqrt[2]*a^(
3/4)*(Sqrt[a]*B - A*Sqrt[c])*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4
)] + 30*Sqrt[2]*a^(3/4)*(-(Sqrt[a]*B) + A*Sqrt[c])*c^(3/4)*ArcTan[1 + (Sqrt[2]*c
^(1/4)*Sqrt[x])/a^(1/4)] + 15*Sqrt[2]*(a^(5/4)*B + a^(3/4)*A*Sqrt[c])*c^(3/4)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 15*Sqrt[2]*(a^(5/4)*B
+ a^(3/4)*A*Sqrt[c])*c^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sq
rt[c]*x])/(60*a^3)

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Maple [A]  time = 0.016, size = 302, normalized size = 1. \[ -{\frac{2\,A}{5\,a}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{3\,a}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Ac}{{a}^{2}\sqrt{x}}}-{\frac{Bc\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{Bc\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{Bc\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{Ac\sqrt{2}}{4\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(c*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319005, size = 1165, normalized size = 3.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*x^(7/2)),x, algorithm="fricas")

[Out]

1/30*(15*a^2*x^(5/2)*sqrt((a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a
^9) + 2*A*B*c^2)/a^4)*log(-(B^4*a^2*c^2 - A^4*c^4)*sqrt(x) + (A*a^7*sqrt(-(B^4*a
^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + B^3*a^4*c - A^2*B*a^3*c^2)*sqrt((a^4*
sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + 2*A*B*c^2)/a^4)) - 15*a^2
*x^(5/2)*sqrt((a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + 2*A*B*
c^2)/a^4)*log(-(B^4*a^2*c^2 - A^4*c^4)*sqrt(x) - (A*a^7*sqrt(-(B^4*a^2*c^3 - 2*A
^2*B^2*a*c^4 + A^4*c^5)/a^9) + B^3*a^4*c - A^2*B*a^3*c^2)*sqrt((a^4*sqrt(-(B^4*a
^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) + 2*A*B*c^2)/a^4)) - 15*a^2*x^(5/2)*sqr
t(-(a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) - 2*A*B*c^2)/a^4)*l
og(-(B^4*a^2*c^2 - A^4*c^4)*sqrt(x) + (A*a^7*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^
4 + A^4*c^5)/a^9) - B^3*a^4*c + A^2*B*a^3*c^2)*sqrt(-(a^4*sqrt(-(B^4*a^2*c^3 - 2
*A^2*B^2*a*c^4 + A^4*c^5)/a^9) - 2*A*B*c^2)/a^4)) + 15*a^2*x^(5/2)*sqrt(-(a^4*sq
rt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^5)/a^9) - 2*A*B*c^2)/a^4)*log(-(B^4*a
^2*c^2 - A^4*c^4)*sqrt(x) - (A*a^7*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a*c^4 + A^4*c^
5)/a^9) - B^3*a^4*c + A^2*B*a^3*c^2)*sqrt(-(a^4*sqrt(-(B^4*a^2*c^3 - 2*A^2*B^2*a
*c^4 + A^4*c^5)/a^9) - 2*A*B*c^2)/a^4)) + 60*A*c*x^2 - 20*B*a*x - 12*A*a)/(a^2*x
^(5/2))

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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**(7/2)/(c*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285546, size = 358, normalized size = 1.23 \[ -\frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{3} c} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c - \left (a c^{3}\right )^{\frac{3}{4}} A\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{3} c} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{3} c} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} B a c + \left (a c^{3}\right )^{\frac{3}{4}} A\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{3} c} + \frac{2 \,{\left (15 \, A c x^{2} - 5 \, B a x - 3 \, A a\right )}}{15 \, a^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*x^(7/2)),x, algorithm="giac")

[Out]

-1/2*sqrt(2)*((a*c^3)^(1/4)*B*a*c - (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^3*c) - 1/2*sqrt(2)*((a*c^3)^(1/4)*B*a*
c - (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^3*c) - 1/4*sqrt(2)*((a*c^3)^(1/4)*B*a*c + (a*c^3)^(3/4)*A)*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)*B*a*c
+ (a*c^3)^(3/4)*A)*ln(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^3*c) + 2
/15*(15*A*c*x^2 - 5*B*a*x - 3*A*a)/(a^2*x^(5/2))