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

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

[Out]

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

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Rubi [A]  time = 0.207055, 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 \[ \frac{7 a^{3/2} (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{11/2}}-\frac{7 a \sqrt{x} (5 A b-9 a B)}{4 b^5}+\frac{7 x^{3/2} (5 A b-9 a B)}{12 b^4}-\frac{7 x^{5/2} (5 A b-9 a B)}{20 a b^3}+\frac{x^{7/2} (5 A b-9 a B)}{4 a b^2 (a+b x)}+\frac{x^{9/2} (A b-a B)}{2 a b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.7531, size = 160, normalized size = 0.95 \[ \frac{7 a^{\frac{3}{2}} \left (5 A b - 9 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{11}{2}}} - \frac{7 a \sqrt{x} \left (5 A b - 9 B a\right )}{4 b^{5}} + \frac{7 x^{\frac{3}{2}} \left (5 A b - 9 B a\right )}{12 b^{4}} + \frac{x^{\frac{9}{2}} \left (A b - B a\right )}{2 a b \left (a + b x\right )^{2}} + \frac{x^{\frac{7}{2}} \left (5 A b - 9 B a\right )}{4 a b^{2} \left (a + b x\right )} - \frac{7 x^{\frac{5}{2}} \left (5 A b - 9 B a\right )}{20 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.198916, size = 129, normalized size = 0.76 \[ \frac{\sqrt{x} \left (945 a^4 B-525 a^3 b (A-3 B x)+7 a^2 b^2 x (72 B x-125 A)-8 a b^3 x^2 (35 A+9 B x)+8 b^4 x^3 (5 A+3 B x)\right )}{60 b^5 (a+b x)^2}-\frac{7 a^{3/2} (9 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 178, normalized size = 1.1 \[{\frac{2\,B}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-2\,{\frac{B{x}^{3/2}a}{{b}^{4}}}-6\,{\frac{aA\sqrt{x}}{{b}^{4}}}+12\,{\frac{B{a}^{2}\sqrt{x}}{{b}^{5}}}-{\frac{13\,A{a}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{17\,B{a}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{11\,A{a}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{15\,B{a}^{4}}{4\,{b}^{5} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{35\,A{a}^{2}}{4\,{b}^{4}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,B{a}^{3}}{4\,{b}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220675, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{120 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac{105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{60 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/120*(105*(9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 2*(9*B*a^3*b
- 5*A*a^2*b^2)*x)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a))
- 2*(24*B*b^4*x^4 + 945*B*a^4 - 525*A*a^3*b - 8*(9*B*a*b^3 - 5*A*b^4)*x^3 + 56*(
9*B*a^2*b^2 - 5*A*a*b^3)*x^2 + 175*(9*B*a^3*b - 5*A*a^2*b^2)*x)*sqrt(x))/(b^7*x^
2 + 2*a*b^6*x + a^2*b^5), -1/60*(105*(9*B*a^4 - 5*A*a^3*b + (9*B*a^2*b^2 - 5*A*a
*b^3)*x^2 + 2*(9*B*a^3*b - 5*A*a^2*b^2)*x)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) -
(24*B*b^4*x^4 + 945*B*a^4 - 525*A*a^3*b - 8*(9*B*a*b^3 - 5*A*b^4)*x^3 + 56*(9*B
*a^2*b^2 - 5*A*a*b^3)*x^2 + 175*(9*B*a^3*b - 5*A*a^2*b^2)*x)*sqrt(x))/(b^7*x^2 +
2*a*b^6*x + a^2*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214186, size = 197, normalized size = 1.17 \[ -\frac{7 \,{\left (9 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{5}} + \frac{17 \, B a^{3} b x^{\frac{3}{2}} - 13 \, A a^{2} b^{2} x^{\frac{3}{2}} + 15 \, B a^{4} \sqrt{x} - 11 \, A a^{3} b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{5}} + \frac{2 \,{\left (3 \, B b^{12} x^{\frac{5}{2}} - 15 \, B a b^{11} x^{\frac{3}{2}} + 5 \, A b^{12} x^{\frac{3}{2}} + 90 \, B a^{2} b^{10} \sqrt{x} - 45 \, A a b^{11} \sqrt{x}\right )}}{15 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/4*(9*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/4*(17
*B*a^3*b*x^(3/2) - 13*A*a^2*b^2*x^(3/2) + 15*B*a^4*sqrt(x) - 11*A*a^3*b*sqrt(x))
/((b*x + a)^2*b^5) + 2/15*(3*B*b^12*x^(5/2) - 15*B*a*b^11*x^(3/2) + 5*A*b^12*x^(
3/2) + 90*B*a^2*b^10*sqrt(x) - 45*A*a*b^11*sqrt(x))/b^15