3.332 \(\int \frac{A+B x}{x^{9/2} (a+b x)^2} \, dx\)

Optimal. Leaf size=153 \[ \frac{b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}+\frac{b^2 (9 A b-7 a B)}{a^5 \sqrt{x}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{A b-a B}{a b x^{7/2} (a+b x)} \]

[Out]

-(9*A*b - 7*a*B)/(7*a^2*b*x^(7/2)) + (9*A*b - 7*a*B)/(5*a^3*x^(5/2)) - (b*(9*A*b
- 7*a*B))/(3*a^4*x^(3/2)) + (b^2*(9*A*b - 7*a*B))/(a^5*Sqrt[x]) + (A*b - a*B)/(
a*b*x^(7/2)*(a + b*x)) + (b^(5/2)*(9*A*b - 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[
a]])/a^(11/2)

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Rubi [A]  time = 0.208247, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}+\frac{b^2 (9 A b-7 a B)}{a^5 \sqrt{x}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{A b-a B}{a b x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(9*A*b - 7*a*B)/(7*a^2*b*x^(7/2)) + (9*A*b - 7*a*B)/(5*a^3*x^(5/2)) - (b*(9*A*b
- 7*a*B))/(3*a^4*x^(3/2)) + (b^2*(9*A*b - 7*a*B))/(a^5*Sqrt[x]) + (A*b - a*B)/(
a*b*x^(7/2)*(a + b*x)) + (b^(5/2)*(9*A*b - 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[
a]])/a^(11/2)

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Rubi in Sympy [A]  time = 27.0137, size = 143, normalized size = 0.93 \[ \frac{A b - B a}{a b x^{\frac{7}{2}} \left (a + b x\right )} - \frac{9 A b - 7 B a}{7 a^{2} b x^{\frac{7}{2}}} + \frac{9 A b - 7 B a}{5 a^{3} x^{\frac{5}{2}}} - \frac{b \left (9 A b - 7 B a\right )}{3 a^{4} x^{\frac{3}{2}}} + \frac{b^{2} \left (9 A b - 7 B a\right )}{a^{5} \sqrt{x}} + \frac{b^{\frac{5}{2}} \left (9 A b - 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*b - B*a)/(a*b*x**(7/2)*(a + b*x)) - (9*A*b - 7*B*a)/(7*a**2*b*x**(7/2)) + (9*
A*b - 7*B*a)/(5*a**3*x**(5/2)) - b*(9*A*b - 7*B*a)/(3*a**4*x**(3/2)) + b**2*(9*A
*b - 7*B*a)/(a**5*sqrt(x)) + b**(5/2)*(9*A*b - 7*B*a)*atan(sqrt(b)*sqrt(x)/sqrt(
a))/a**(11/2)

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Mathematica [A]  time = 0.171236, size = 131, normalized size = 0.86 \[ \frac{b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}+\frac{-6 a^4 (5 A+7 B x)+2 a^3 b x (27 A+49 B x)-14 a^2 b^2 x^2 (9 A+35 B x)+105 a b^3 x^3 (6 A-7 B x)+945 A b^4 x^4}{105 a^5 x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(945*A*b^4*x^4 + 105*a*b^3*x^3*(6*A - 7*B*x) - 6*a^4*(5*A + 7*B*x) - 14*a^2*b^2*
x^2*(9*A + 35*B*x) + 2*a^3*b*x*(27*A + 49*B*x))/(105*a^5*x^(7/2)*(a + b*x)) + (b
^(5/2)*(9*A*b - 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(11/2)

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Maple [A]  time = 0.027, size = 163, normalized size = 1.1 \[ -{\frac{2\,A}{7\,{a}^{2}}{x}^{-{\frac{7}{2}}}}+{\frac{4\,Ab}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}-2\,{\frac{A{b}^{2}}{{a}^{4}{x}^{3/2}}}+{\frac{4\,Bb}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+8\,{\frac{A{b}^{3}}{{a}^{5}\sqrt{x}}}-6\,{\frac{B{b}^{2}}{{a}^{4}\sqrt{x}}}+{\frac{{b}^{4}A}{{a}^{5} \left ( bx+a \right ) }\sqrt{x}}-{\frac{{b}^{3}B}{{a}^{4} \left ( bx+a \right ) }\sqrt{x}}+9\,{\frac{{b}^{4}A}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-7\,{\frac{{b}^{3}B}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*A/a^2/x^(7/2)+4/5/a^3/x^(5/2)*A*b-2/5/a^2/x^(5/2)*B-2*b^2/a^4/x^(3/2)*A+4/3
*b/a^3/x^(3/2)*B+8*b^3/a^5/x^(1/2)*A-6*b^2/a^4/x^(1/2)*B+1/a^5*b^4*x^(1/2)/(b*x+
a)*A-1/a^4*b^3*x^(1/2)/(b*x+a)*B+9/a^5*b^4/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1
/2))*A-7/a^4*b^3/(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)^2*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222839, size = 1, normalized size = 0.01 \[ \left [-\frac{60 \, A a^{4} + 210 \,{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 140 \,{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 28 \,{\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 105 \,{\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} +{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3}\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 12 \,{\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x}{210 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )} \sqrt{x}}, -\frac{30 \, A a^{4} + 105 \,{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \,{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \,{\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} - 105 \,{\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} +{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3}\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 6 \,{\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x}{105 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/210*(60*A*a^4 + 210*(7*B*a*b^3 - 9*A*b^4)*x^4 + 140*(7*B*a^2*b^2 - 9*A*a*b^3
)*x^3 - 28*(7*B*a^3*b - 9*A*a^2*b^2)*x^2 + 105*((7*B*a*b^3 - 9*A*b^4)*x^4 + (7*B
*a^2*b^2 - 9*A*a*b^3)*x^3)*sqrt(x)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a)
- a)/(b*x + a)) + 12*(7*B*a^4 - 9*A*a^3*b)*x)/((a^5*b*x^4 + a^6*x^3)*sqrt(x)), -
1/105*(30*A*a^4 + 105*(7*B*a*b^3 - 9*A*b^4)*x^4 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)*x
^3 - 14*(7*B*a^3*b - 9*A*a^2*b^2)*x^2 - 105*((7*B*a*b^3 - 9*A*b^4)*x^4 + (7*B*a^
2*b^2 - 9*A*a*b^3)*x^3)*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 6*(7
*B*a^4 - 9*A*a^3*b)*x)/((a^5*b*x^4 + a^6*x^3)*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**(9/2)/(b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217963, size = 184, normalized size = 1.2 \[ -\frac{{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} - \frac{B a b^{3} \sqrt{x} - A b^{4} \sqrt{x}}{{\left (b x + a\right )} a^{5}} - \frac{2 \,{\left (315 \, B a b^{2} x^{3} - 420 \, A b^{3} x^{3} - 70 \, B a^{2} b x^{2} + 105 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 42 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{5} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(7*B*a*b^3 - 9*A*b^4)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) - (B*a*b^3*sq
rt(x) - A*b^4*sqrt(x))/((b*x + a)*a^5) - 2/105*(315*B*a*b^2*x^3 - 420*A*b^3*x^3
- 70*B*a^2*b*x^2 + 105*A*a*b^2*x^2 + 21*B*a^3*x - 42*A*a^2*b*x + 15*A*a^3)/(a^5*
x^(7/2))