3.340 \(\int \frac{A+B x}{x^{7/2} (a+b x)^3} \, dx\)

Optimal. Leaf size=169 \[ -\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{11/2}}-\frac{7 b (9 A b-5 a B)}{4 a^5 \sqrt{x}}+\frac{7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac{7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac{9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}+\frac{A b-a B}{2 a b x^{5/2} (a+b x)^2} \]

[Out]

(-7*(9*A*b - 5*a*B))/(20*a^3*b*x^(5/2)) + (7*(9*A*b - 5*a*B))/(12*a^4*x^(3/2)) -
(7*b*(9*A*b - 5*a*B))/(4*a^5*Sqrt[x]) + (A*b - a*B)/(2*a*b*x^(5/2)*(a + b*x)^2)
+ (9*A*b - 5*a*B)/(4*a^2*b*x^(5/2)*(a + b*x)) - (7*b^(3/2)*(9*A*b - 5*a*B)*ArcT
an[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(11/2))

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Rubi [A]  time = 0.213976, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{7 b^{3/2} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{11/2}}-\frac{7 b (9 A b-5 a B)}{4 a^5 \sqrt{x}}+\frac{7 (9 A b-5 a B)}{12 a^4 x^{3/2}}-\frac{7 (9 A b-5 a B)}{20 a^3 b x^{5/2}}+\frac{9 A b-5 a B}{4 a^2 b x^{5/2} (a+b x)}+\frac{A b-a B}{2 a b x^{5/2} (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*(a + b*x)^3),x]

[Out]

(-7*(9*A*b - 5*a*B))/(20*a^3*b*x^(5/2)) + (7*(9*A*b - 5*a*B))/(12*a^4*x^(3/2)) -
(7*b*(9*A*b - 5*a*B))/(4*a^5*Sqrt[x]) + (A*b - a*B)/(2*a*b*x^(5/2)*(a + b*x)^2)
+ (9*A*b - 5*a*B)/(4*a^2*b*x^(5/2)*(a + b*x)) - (7*b^(3/2)*(9*A*b - 5*a*B)*ArcT
an[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(11/2))

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Rubi in Sympy [A]  time = 26.3772, size = 156, normalized size = 0.92 \[ \frac{A b - B a}{2 a b x^{\frac{5}{2}} \left (a + b x\right )^{2}} + \frac{9 A b - 5 B a}{4 a^{2} b x^{\frac{5}{2}} \left (a + b x\right )} - \frac{7 \left (9 A b - 5 B a\right )}{20 a^{3} b x^{\frac{5}{2}}} + \frac{7 \left (9 A b - 5 B a\right )}{12 a^{4} x^{\frac{3}{2}}} - \frac{7 b \left (9 A b - 5 B a\right )}{4 a^{5} \sqrt{x}} - \frac{7 b^{\frac{3}{2}} \left (9 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(7/2)/(b*x+a)**3,x)

[Out]

(A*b - B*a)/(2*a*b*x**(5/2)*(a + b*x)**2) + (9*A*b - 5*B*a)/(4*a**2*b*x**(5/2)*(
a + b*x)) - 7*(9*A*b - 5*B*a)/(20*a**3*b*x**(5/2)) + 7*(9*A*b - 5*B*a)/(12*a**4*
x**(3/2)) - 7*b*(9*A*b - 5*B*a)/(4*a**5*sqrt(x)) - 7*b**(3/2)*(9*A*b - 5*B*a)*at
an(sqrt(b)*sqrt(x)/sqrt(a))/(4*a**(11/2))

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Mathematica [A]  time = 0.192197, size = 133, normalized size = 0.79 \[ \frac{7 b^{3/2} (5 a B-9 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{11/2}}+\frac{-8 a^4 (3 A+5 B x)+8 a^3 b x (9 A+35 B x)+7 a^2 b^2 x^2 (125 B x-72 A)+525 a b^3 x^3 (B x-3 A)-945 A b^4 x^4}{60 a^5 x^{5/2} (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*(a + b*x)^3),x]

[Out]

(-945*A*b^4*x^4 + 525*a*b^3*x^3*(-3*A + B*x) - 8*a^4*(3*A + 5*B*x) + 8*a^3*b*x*(
9*A + 35*B*x) + 7*a^2*b^2*x^2*(-72*A + 125*B*x))/(60*a^5*x^(5/2)*(a + b*x)^2) +
(7*b^(3/2)*(-9*A*b + 5*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(11/2))

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Maple [A]  time = 0.027, size = 178, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}+2\,{\frac{Ab}{{a}^{4}{x}^{3/2}}}-{\frac{2\,B}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-12\,{\frac{{b}^{2}A}{{a}^{5}\sqrt{x}}}+6\,{\frac{Bb}{{a}^{4}\sqrt{x}}}-{\frac{15\,{b}^{4}A}{4\,{a}^{5} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{11\,{b}^{3}B}{4\,{a}^{4} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,{b}^{3}A}{4\,{a}^{4} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{13\,{b}^{2}B}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{63\,{b}^{3}A}{4\,{a}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{b}^{2}B}{4\,{a}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(b*x+a)^3,x)

[Out]

-2/5*A/a^3/x^(5/2)+2/a^4/x^(3/2)*A*b-2/3/a^3/x^(3/2)*B-12*b^2/a^5/x^(1/2)*A+6*b/
a^4/x^(1/2)*B-15/4/a^5*b^4/(b*x+a)^2*x^(3/2)*A+11/4/a^4*b^3/(b*x+a)^2*x^(3/2)*B-
17/4/a^4*b^3/(b*x+a)^2*A*x^(1/2)+13/4/a^3*b^2/(b*x+a)^2*B*x^(1/2)-63/4/a^5*b^3/(
a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+35/4/a^4*b^2/(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^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223936, size = 1, normalized size = 0.01 \[ \left [-\frac{48 \, A a^{4} - 210 \,{\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 350 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 112 \,{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 105 \,{\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 2 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} +{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 16 \,{\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x}{120 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )} \sqrt{x}}, -\frac{24 \, A a^{4} - 105 \,{\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 175 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \,{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 105 \,{\left ({\left (5 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 2 \,{\left (5 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} +{\left (5 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 8 \,{\left (5 \, B a^{4} - 9 \, A a^{3} b\right )} x}{60 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*x^(7/2)),x, algorithm="fricas")

[Out]

[-1/120*(48*A*a^4 - 210*(5*B*a*b^3 - 9*A*b^4)*x^4 - 350*(5*B*a^2*b^2 - 9*A*a*b^3
)*x^3 - 112*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 105*((5*B*a*b^3 - 9*A*b^4)*x^4 + 2*(
5*B*a^2*b^2 - 9*A*a*b^3)*x^3 + (5*B*a^3*b - 9*A*a^2*b^2)*x^2)*sqrt(x)*sqrt(-b/a)
*log((b*x - 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 16*(5*B*a^4 - 9*A*a^3*b)*x)
/((a^5*b^2*x^4 + 2*a^6*b*x^3 + a^7*x^2)*sqrt(x)), -1/60*(24*A*a^4 - 105*(5*B*a*b
^3 - 9*A*b^4)*x^4 - 175*(5*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*
b^2)*x^2 + 105*((5*B*a*b^3 - 9*A*b^4)*x^4 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*x^3 + (5
*B*a^3*b - 9*A*a^2*b^2)*x^2)*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) +
8*(5*B*a^4 - 9*A*a^3*b)*x)/((a^5*b^2*x^4 + 2*a^6*b*x^3 + a^7*x^2)*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**(7/2)/(b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215497, size = 182, normalized size = 1.08 \[ \frac{7 \,{\left (5 \, B a b^{2} - 9 \, A b^{3}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{5}} + \frac{11 \, B a b^{3} x^{\frac{3}{2}} - 15 \, A b^{4} x^{\frac{3}{2}} + 13 \, B a^{2} b^{2} \sqrt{x} - 17 \, A a b^{3} \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{5}} + \frac{2 \,{\left (45 \, B a b x^{2} - 90 \, A b^{2} x^{2} - 5 \, B a^{2} x + 15 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{5} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*x^(7/2)),x, algorithm="giac")

[Out]

7/4*(5*B*a*b^2 - 9*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) + 1/4*(11*
B*a*b^3*x^(3/2) - 15*A*b^4*x^(3/2) + 13*B*a^2*b^2*sqrt(x) - 17*A*a*b^3*sqrt(x))/
((b*x + a)^2*a^5) + 2/15*(45*B*a*b*x^2 - 90*A*b^2*x^2 - 5*B*a^2*x + 15*A*a*b*x -
3*A*a^2)/(a^5*x^(5/2))