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

Optimal. Leaf size=130 \[ \frac{a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}-\frac{a \sqrt{x} (5 A b-7 a B)}{b^4}+\frac{x^{3/2} (5 A b-7 a B)}{3 b^3}-\frac{x^{5/2} (5 A b-7 a B)}{5 a b^2}+\frac{x^{7/2} (A b-a B)}{a b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.167313, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}-\frac{a \sqrt{x} (5 A b-7 a B)}{b^4}+\frac{x^{3/2} (5 A b-7 a B)}{3 b^3}-\frac{x^{5/2} (5 A b-7 a B)}{5 a b^2}+\frac{x^{7/2} (A b-a B)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.1689, size = 119, normalized size = 0.92 \[ \frac{a^{\frac{3}{2}} \left (5 A b - 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{9}{2}}} - \frac{a \sqrt{x} \left (5 A b - 7 B a\right )}{b^{4}} + \frac{x^{\frac{3}{2}} \left (5 A b - 7 B a\right )}{3 b^{3}} + \frac{x^{\frac{7}{2}} \left (A b - B a\right )}{a b \left (a + b x\right )} - \frac{x^{\frac{5}{2}} \left (5 A b - 7 B a\right )}{5 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.147406, size = 110, normalized size = 0.85 \[ \frac{\sqrt{x} \left (105 a^3 B+a^2 (70 b B x-75 A b)-2 a b^2 x (25 A+7 B x)+2 b^3 x^2 (5 A+3 B x)\right )}{15 b^4 (a+b x)}-\frac{a^{3/2} (7 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(105*a^3*B + 2*b^3*x^2*(5*A + 3*B*x) - 2*a*b^2*x*(25*A + 7*B*x) + a^2*(
-75*A*b + 70*b*B*x)))/(15*b^4*(a + b*x)) - (a^(3/2)*(-5*A*b + 7*a*B)*ArcTan[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/b^(9/2)

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Maple [A]  time = 0.019, size = 139, normalized size = 1.1 \[{\frac{2\,B}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{4\,Ba}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-4\,{\frac{aA\sqrt{x}}{{b}^{3}}}+6\,{\frac{B{a}^{2}\sqrt{x}}{{b}^{4}}}-{\frac{A{a}^{2}}{{b}^{3} \left ( bx+a \right ) }\sqrt{x}}+{\frac{B{a}^{3}}{{b}^{4} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{A{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-7\,{\frac{B{a}^{3}}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232353, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \,{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{x}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \,{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{x}}{15 \,{\left (b^{5} x + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/30*(15*(7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x)*sqrt(-a/b)*log((b*x
+ 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(6*B*b^3*x^3 + 105*B*a^3 - 75*A*a^
2*b - 2*(7*B*a*b^2 - 5*A*b^3)*x^2 + 10*(7*B*a^2*b - 5*A*a*b^2)*x)*sqrt(x))/(b^5*
x + a*b^4), -1/15*(15*(7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x)*sqrt(a/b
)*arctan(sqrt(x)/sqrt(a/b)) - (6*B*b^3*x^3 + 105*B*a^3 - 75*A*a^2*b - 2*(7*B*a*b
^2 - 5*A*b^3)*x^2 + 10*(7*B*a^2*b - 5*A*a*b^2)*x)*sqrt(x))/(b^5*x + a*b^4)]

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Sympy [A]  time = 150.419, size = 563, normalized size = 4.33 \[ A \left (\frac{15 a^{\frac{61}{2}} b^{17} x^{\frac{41}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} + \frac{15 a^{\frac{59}{2}} b^{18} x^{\frac{43}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} - \frac{15 a^{30} b^{\frac{35}{2}} x^{21}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} - \frac{10 a^{29} b^{\frac{37}{2}} x^{22}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}} + \frac{2 a^{28} b^{\frac{39}{2}} x^{23}}{3 a^{29} b^{\frac{41}{2}} x^{\frac{41}{2}} + 3 a^{28} b^{\frac{43}{2}} x^{\frac{43}{2}}}\right ) + B \left (- \frac{105 a^{\frac{121}{2}} b^{30} x^{\frac{69}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{15 a^{58} b^{\frac{69}{2}} x^{\frac{69}{2}} + 15 a^{57} b^{\frac{71}{2}} x^{\frac{71}{2}}} - \frac{105 a^{\frac{119}{2}} b^{31} x^{\frac{71}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{15 a^{58} b^{\frac{69}{2}} x^{\frac{69}{2}} + 15 a^{57} b^{\frac{71}{2}} x^{\frac{71}{2}}} + \frac{105 a^{60} b^{\frac{61}{2}} x^{35}}{15 a^{58} b^{\frac{69}{2}} x^{\frac{69}{2}} + 15 a^{57} b^{\frac{71}{2}} x^{\frac{71}{2}}} + \frac{70 a^{59} b^{\frac{63}{2}} x^{36}}{15 a^{58} b^{\frac{69}{2}} x^{\frac{69}{2}} + 15 a^{57} b^{\frac{71}{2}} x^{\frac{71}{2}}} - \frac{14 a^{58} b^{\frac{65}{2}} x^{37}}{15 a^{58} b^{\frac{69}{2}} x^{\frac{69}{2}} + 15 a^{57} b^{\frac{71}{2}} x^{\frac{71}{2}}} + \frac{6 a^{57} b^{\frac{67}{2}} x^{38}}{15 a^{58} b^{\frac{69}{2}} x^{\frac{69}{2}} + 15 a^{57} b^{\frac{71}{2}} x^{\frac{71}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(15*a**(61/2)*b**17*x**(41/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**29*b**(41/2)
*x**(41/2) + 3*a**28*b**(43/2)*x**(43/2)) + 15*a**(59/2)*b**18*x**(43/2)*atan(sq
rt(b)*sqrt(x)/sqrt(a))/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x**(43/2
)) - 15*a**30*b**(35/2)*x**21/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(43/2)*x
**(43/2)) - 10*a**29*b**(37/2)*x**22/(3*a**29*b**(41/2)*x**(41/2) + 3*a**28*b**(
43/2)*x**(43/2)) + 2*a**28*b**(39/2)*x**23/(3*a**29*b**(41/2)*x**(41/2) + 3*a**2
8*b**(43/2)*x**(43/2))) + B*(-105*a**(121/2)*b**30*x**(69/2)*atan(sqrt(b)*sqrt(x
)/sqrt(a))/(15*a**58*b**(69/2)*x**(69/2) + 15*a**57*b**(71/2)*x**(71/2)) - 105*a
**(119/2)*b**31*x**(71/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(15*a**58*b**(69/2)*x**(
69/2) + 15*a**57*b**(71/2)*x**(71/2)) + 105*a**60*b**(61/2)*x**35/(15*a**58*b**(
69/2)*x**(69/2) + 15*a**57*b**(71/2)*x**(71/2)) + 70*a**59*b**(63/2)*x**36/(15*a
**58*b**(69/2)*x**(69/2) + 15*a**57*b**(71/2)*x**(71/2)) - 14*a**58*b**(65/2)*x*
*37/(15*a**58*b**(69/2)*x**(69/2) + 15*a**57*b**(71/2)*x**(71/2)) + 6*a**57*b**(
67/2)*x**38/(15*a**58*b**(69/2)*x**(69/2) + 15*a**57*b**(71/2)*x**(71/2)))

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GIAC/XCAS [A]  time = 0.212025, size = 165, normalized size = 1.27 \[ -\frac{{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{B a^{3} \sqrt{x} - A a^{2} b \sqrt{x}}{{\left (b x + a\right )} b^{4}} + \frac{2 \,{\left (3 \, B b^{8} x^{\frac{5}{2}} - 10 \, B a b^{7} x^{\frac{3}{2}} + 5 \, A b^{8} x^{\frac{3}{2}} + 45 \, B a^{2} b^{6} \sqrt{x} - 30 \, A a b^{7} \sqrt{x}\right )}}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(7*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + (B*a^3*sqrt
(x) - A*a^2*b*sqrt(x))/((b*x + a)*b^4) + 2/15*(3*B*b^8*x^(5/2) - 10*B*a*b^7*x^(3
/2) + 5*A*b^8*x^(3/2) + 45*B*a^2*b^6*sqrt(x) - 30*A*a*b^7*sqrt(x))/b^10