3.339 \(\int \frac{A+B x}{x^{5/2} (a+b x)^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{5 (7 A b-3 a B)}{4 a^4 \sqrt{x}}-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2} \]

[Out]

(-5*(7*A*b - 3*a*B))/(12*a^3*b*x^(3/2)) + (5*(7*A*b - 3*a*B))/(4*a^4*Sqrt[x]) +
(A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x)^2) + (7*A*b - 3*a*B)/(4*a^2*b*x^(3/2)*(a +
b*x)) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2)
)

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Rubi [A]  time = 0.178685, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{5 (7 A b-3 a B)}{4 a^4 \sqrt{x}}-\frac{5 (7 A b-3 a B)}{12 a^3 b x^{3/2}}+\frac{7 A b-3 a B}{4 a^2 b x^{3/2} (a+b x)}+\frac{A b-a B}{2 a b x^{3/2} (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(5/2)*(a + b*x)^3),x]

[Out]

(-5*(7*A*b - 3*a*B))/(12*a^3*b*x^(3/2)) + (5*(7*A*b - 3*a*B))/(4*a^4*Sqrt[x]) +
(A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x)^2) + (7*A*b - 3*a*B)/(4*a^2*b*x^(3/2)*(a +
b*x)) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2)
)

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Rubi in Sympy [A]  time = 21.3654, size = 133, normalized size = 0.9 \[ \frac{A b - B a}{2 a b x^{\frac{3}{2}} \left (a + b x\right )^{2}} + \frac{7 A b - 3 B a}{4 a^{2} b x^{\frac{3}{2}} \left (a + b x\right )} - \frac{5 \left (7 A b - 3 B a\right )}{12 a^{3} b x^{\frac{3}{2}}} + \frac{5 \left (7 A b - 3 B a\right )}{4 a^{4} \sqrt{x}} + \frac{5 \sqrt{b} \left (7 A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(b*x+a)**3,x)

[Out]

(A*b - B*a)/(2*a*b*x**(3/2)*(a + b*x)**2) + (7*A*b - 3*B*a)/(4*a**2*b*x**(3/2)*(
a + b*x)) - 5*(7*A*b - 3*B*a)/(12*a**3*b*x**(3/2)) + 5*(7*A*b - 3*B*a)/(4*a**4*s
qrt(x)) + 5*sqrt(b)*(7*A*b - 3*B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(9/2))

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Mathematica [A]  time = 0.17563, size = 112, normalized size = 0.76 \[ \frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{-8 a^3 (A+3 B x)+a^2 b x (56 A-75 B x)+5 a b^2 x^2 (35 A-9 B x)+105 A b^3 x^3}{12 a^4 x^{3/2} (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(5/2)*(a + b*x)^3),x]

[Out]

(105*A*b^3*x^3 + a^2*b*x*(56*A - 75*B*x) + 5*a*b^2*x^2*(35*A - 9*B*x) - 8*a^3*(A
+ 3*B*x))/(12*a^4*x^(3/2)*(a + b*x)^2) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTan[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))

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Maple [A]  time = 0.027, size = 152, normalized size = 1. \[ -{\frac{2\,A}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+6\,{\frac{Ab}{\sqrt{x}{a}^{4}}}-2\,{\frac{B}{\sqrt{x}{a}^{3}}}+{\frac{11\,{b}^{3}A}{4\,{a}^{4} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{7\,{b}^{2}B}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{13\,{b}^{2}A}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{9\,Bb}{4\,{a}^{2} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{35\,{b}^{2}A}{4\,{a}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,Bb}{4\,{a}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(b*x+a)^3,x)

[Out]

-2/3*A/a^3/x^(3/2)+6/x^(1/2)/a^4*A*b-2/x^(1/2)/a^3*B+11/4/a^4*b^3/(b*x+a)^2*x^(3
/2)*A-7/4/a^3*b^2/(b*x+a)^2*x^(3/2)*B+13/4/a^3*b^2/(b*x+a)^2*A*x^(1/2)-9/4/a^2*b
/(b*x+a)^2*B*x^(1/2)+35/4/a^4*b^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-15
/4/a^3*b/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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)/((b*x + a)^3*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226128, size = 1, normalized size = 0.01 \[ \left [-\frac{16 \, A a^{3} + 30 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 50 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 16 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x}{24 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )} \sqrt{x}}, -\frac{8 \, A a^{3} + 15 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} - 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 8 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x}{12 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/24*(16*A*a^3 + 30*(3*B*a*b^2 - 7*A*b^3)*x^3 + 50*(3*B*a^2*b - 7*A*a*b^2)*x^2
+ 15*((3*B*a*b^2 - 7*A*b^3)*x^3 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^2 + (3*B*a^3 - 7*
A*a^2*b)*x)*sqrt(x)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a))
+ 16*(3*B*a^3 - 7*A*a^2*b)*x)/((a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)*sqrt(x)), -1
/12*(8*A*a^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^2 - 1
5*((3*B*a*b^2 - 7*A*b^3)*x^3 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^2 + (3*B*a^3 - 7*A*a^
2*b)*x)*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 8*(3*B*a^3 - 7*A*a^2
*b)*x)/((a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)*sqrt(x))]

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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**(5/2)/(b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216221, size = 146, normalized size = 0.99 \[ -\frac{5 \,{\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{4}} - \frac{2 \,{\left (3 \, B a x - 9 \, A b x + A a\right )}}{3 \, a^{4} x^{\frac{3}{2}}} - \frac{7 \, B a b^{2} x^{\frac{3}{2}} - 11 \, A b^{3} x^{\frac{3}{2}} + 9 \, B a^{2} b \sqrt{x} - 13 \, A a b^{2} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*x^(5/2)),x, algorithm="giac")

[Out]

-5/4*(3*B*a*b - 7*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 2/3*(3*B*
a*x - 9*A*b*x + A*a)/(a^4*x^(3/2)) - 1/4*(7*B*a*b^2*x^(3/2) - 11*A*b^3*x^(3/2) +
9*B*a^2*b*sqrt(x) - 13*A*a*b^2*sqrt(x))/((b*x + a)^2*a^4)