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

Optimal. Leaf size=154 \[ -\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}+\frac{a^2 \sqrt{x} (7 A b-9 a B)}{b^5}-\frac{a x^{3/2} (7 A b-9 a B)}{3 b^4}+\frac{x^{5/2} (7 A b-9 a B)}{5 b^3}-\frac{x^{7/2} (7 A b-9 a B)}{7 a b^2}+\frac{x^{9/2} (A b-a B)}{a b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.206829, antiderivative size = 154, 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 \[ -\frac{a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}+\frac{a^2 \sqrt{x} (7 A b-9 a B)}{b^5}-\frac{a x^{3/2} (7 A b-9 a B)}{3 b^4}+\frac{x^{5/2} (7 A b-9 a B)}{5 b^3}-\frac{x^{7/2} (7 A b-9 a B)}{7 a b^2}+\frac{x^{9/2} (A b-a B)}{a b (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.3781, size = 143, normalized size = 0.93 \[ - \frac{a^{\frac{5}{2}} \left (7 A b - 9 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{11}{2}}} + \frac{a^{2} \sqrt{x} \left (7 A b - 9 B a\right )}{b^{5}} - \frac{a x^{\frac{3}{2}} \left (7 A b - 9 B a\right )}{3 b^{4}} + \frac{x^{\frac{5}{2}} \left (7 A b - 9 B a\right )}{5 b^{3}} + \frac{x^{\frac{9}{2}} \left (A b - B a\right )}{a b \left (a + b x\right )} - \frac{x^{\frac{7}{2}} \left (7 A b - 9 B a\right )}{7 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.173133, size = 128, normalized size = 0.83 \[ \frac{a^{5/2} (9 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}+\frac{\sqrt{x} \left (-945 a^4 B+105 a^3 b (7 A-6 B x)+14 a^2 b^2 x (35 A+9 B x)-2 a b^3 x^2 (49 A+27 B x)+6 b^4 x^3 (7 A+5 B x)\right )}{105 b^5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 163, normalized size = 1.1 \[{\frac{2\,B}{7\,{b}^{2}}{x}^{{\frac{7}{2}}}}+{\frac{2\,A}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}-{\frac{4\,Ba}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}-{\frac{4\,Aa}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}+2\,{\frac{B{x}^{3/2}{a}^{2}}{{b}^{4}}}+6\,{\frac{{a}^{2}A\sqrt{x}}{{b}^{4}}}-8\,{\frac{B{a}^{3}\sqrt{x}}{{b}^{5}}}+{\frac{A{a}^{3}}{{b}^{4} \left ( bx+a \right ) }\sqrt{x}}-{\frac{B{a}^{4}}{{b}^{5} \left ( bx+a \right ) }\sqrt{x}}-7\,{\frac{A{a}^{3}}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+9\,{\frac{B{a}^{4}}{{b}^{5}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23292, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b +{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \,{\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \,{\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \,{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{210 \,{\left (b^{6} x + a b^{5}\right )}}, \frac{105 \,{\left (9 \, B a^{4} - 7 \, A a^{3} b +{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \,{\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \,{\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \,{\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{105 \,{\left (b^{6} x + a b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/210*(105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(-a/b)*log(
(b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(30*B*b^4*x^4 - 945*B*a^4 + 73
5*A*a^3*b - 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 70*
(9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(x))/(b^6*x + a*b^5), 1/105*(105*(9*B*a^4 - 7*A
*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) + (30*
B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*b - 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*
b^2 - 7*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(x))/(b^6*x + a*b^5)]

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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(x**(7/2)*(B*x+A)/(b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219989, size = 197, normalized size = 1.28 \[ \frac{{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} - \frac{B a^{4} \sqrt{x} - A a^{3} b \sqrt{x}}{{\left (b x + a\right )} b^{5}} + \frac{2 \,{\left (15 \, B b^{12} x^{\frac{7}{2}} - 42 \, B a b^{11} x^{\frac{5}{2}} + 21 \, A b^{12} x^{\frac{5}{2}} + 105 \, B a^{2} b^{10} x^{\frac{3}{2}} - 70 \, A a b^{11} x^{\frac{3}{2}} - 420 \, B a^{3} b^{9} \sqrt{x} + 315 \, A a^{2} b^{10} \sqrt{x}\right )}}{105 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(9*B*a^4 - 7*A*a^3*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) - (B*a^4*sqrt(
x) - A*a^3*b*sqrt(x))/((b*x + a)*b^5) + 2/105*(15*B*b^12*x^(7/2) - 42*B*a*b^11*x
^(5/2) + 21*A*b^12*x^(5/2) + 105*B*a^2*b^10*x^(3/2) - 70*A*a*b^11*x^(3/2) - 420*
B*a^3*b^9*sqrt(x) + 315*A*a^2*b^10*sqrt(x))/b^14