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

Optimal. Leaf size=302 \[ \frac{x^{9/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (5 A b-9 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 a \sqrt{x} (a+b x) (5 A b-9 a B)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{3/2} (a+b x) (5 A b-9 a B)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 x^{5/2} (a+b x) (5 A b-9 a B)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 a^{3/2} (a+b x) (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.401912, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{x^{9/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{7/2} (5 A b-9 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 a \sqrt{x} (a+b x) (5 A b-9 a B)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 x^{3/2} (a+b x) (5 A b-9 a B)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 x^{5/2} (a+b x) (5 A b-9 a B)}{20 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 a^{3/2} (a+b x) (5 A b-9 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.18532, size = 148, normalized size = 0.49 \[ \frac{\sqrt{b} \sqrt{x} \left (945 a^4 B-525 a^3 b (A-3 B x)+7 a^2 b^2 x (72 B x-125 A)-8 a b^3 x^2 (35 A+9 B x)+8 b^4 x^3 (5 A+3 B x)\right )-105 a^{3/2} (a+b x)^2 (9 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{60 b^{11/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*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)) - 105*a^(3/2)*(-5*A*
b + 9*a*B)*(a + b*x)^2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(60*b^(11/2)*(a + b*x)
*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.013, size = 283, normalized size = 0.9 \[{\frac{bx+a}{60\,{b}^{5}} \left ( 24\,B\sqrt{ab}{x}^{9/2}{b}^{4}-72\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+40\,A\sqrt{ab}{x}^{7/2}{b}^{4}+504\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}-280\,A\sqrt{ab}{x}^{5/2}a{b}^{3}-875\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}+525\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}+1575\,B\sqrt{ab}{x}^{3/2}{a}^{3}b-945\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}+1050\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}-1890\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{4}b-525\,A\sqrt{ab}\sqrt{x}{a}^{3}b+525\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4}b+945\,B\sqrt{ab}\sqrt{x}{a}^{4}-945\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(24*B*(a*b)^(1/2)*x^(9/2)*b^4-72*B*(a*b)^(1/2)*x^(7/2)*a*b^3+40*A*(a*b)^(1/
2)*x^(7/2)*b^4+504*B*(a*b)^(1/2)*x^(5/2)*a^2*b^2-280*A*(a*b)^(1/2)*x^(5/2)*a*b^3
-875*A*(a*b)^(1/2)*x^(3/2)*a^2*b^2+525*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^2*b
^3+1575*B*(a*b)^(1/2)*x^(3/2)*a^3*b-945*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^3*
b^2+1050*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^3*b^2-1890*B*arctan(x^(1/2)*b/(a*b)
^(1/2))*x*a^4*b-525*A*(a*b)^(1/2)*x^(1/2)*a^3*b+525*A*arctan(x^(1/2)*b/(a*b)^(1/
2))*a^4*b+945*B*(a*b)^(1/2)*x^(1/2)*a^4-945*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)
*(b*x+a)/(a*b)^(1/2)/b^5/((b*x+a)^2)^(3/2)

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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^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288817, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{120 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac{105 \,{\left (9 \, B a^{4} - 5 \, A a^{3} b +{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 2 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (24 \, B b^{4} x^{4} + 945 \, B a^{4} - 525 \, A a^{3} b - 8 \,{\left (9 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 56 \,{\left (9 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} + 175 \,{\left (9 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{60 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280374, size = 230, normalized size = 0.76 \[ -\frac{7 \,{\left (9 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{5}{\rm sign}\left (b x + a\right )} + \frac{17 \, B a^{3} b x^{\frac{3}{2}} - 13 \, A a^{2} b^{2} x^{\frac{3}{2}} + 15 \, B a^{4} \sqrt{x} - 11 \, A a^{3} b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{5}{\rm sign}\left (b x + a\right )} + \frac{2 \,{\left (3 \, B b^{12} x^{\frac{5}{2}} - 15 \, B a b^{11} x^{\frac{3}{2}} + 5 \, A b^{12} x^{\frac{3}{2}} + 90 \, B a^{2} b^{10} \sqrt{x} - 45 \, A a b^{11} \sqrt{x}\right )}}{15 \, b^{15}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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