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

Optimal. Leaf size=286 \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.375064, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{2 x^{7/2} (a+b x) (A b-a B)}{7 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 x^{3/2} (a+b x) (A b-a B)}{3 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{5/2} (a+b x) (A b-a B)}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{9/2} (a+b x)}{9 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{7/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^3 \sqrt{x} (a+b x) (A b-a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^3*(A*b - a*B)*Sqrt[x]*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*
a^2*(A*b - a*B)*x^(3/2)*(a + b*x))/(3*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*a*
(A*b - a*B)*x^(5/2)*(a + b*x))/(5*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*(A*b -
a*B)*x^(7/2)*(a + b*x))/(7*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*B*x^(9/2)*(a
+ b*x))/(9*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^(7/2)*(A*b - a*B)*(a + b*x)*
ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(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*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.17593, size = 139, normalized size = 0.49 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} \left (315 a^4 B-105 a^3 b (3 A+B x)+21 a^2 b^2 x (5 A+3 B x)-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )-315 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{315 b^{11/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[x]*(315*a^4*B - 105*a^3*b*(3*A + B*x) + 21*a^2*b^2*x*
(5*A + 3*B*x) - 9*a*b^3*x^2*(7*A + 5*B*x) + 5*b^4*x^3*(9*A + 7*B*x)) - 315*a^(7/
2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(315*b^(11/2)*Sqrt[(a + b*
x)^2])

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Maple [A]  time = 0.013, size = 197, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{315\,{b}^{5}} \left ( 35\,B\sqrt{ab}{x}^{9/2}{b}^{4}+45\,A\sqrt{ab}{x}^{7/2}{b}^{4}-45\,B\sqrt{ab}{x}^{7/2}a{b}^{3}-63\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+63\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+105\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}-105\,B\sqrt{ab}{x}^{3/2}{a}^{3}b-315\,A\sqrt{ab}\sqrt{x}{a}^{3}b+315\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4}b+315\,B\sqrt{ab}\sqrt{x}{a}^{4}-315\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)/((b*x+a)^2)^(1/2),x)

[Out]

2/315*(b*x+a)*(35*B*(a*b)^(1/2)*x^(9/2)*b^4+45*A*(a*b)^(1/2)*x^(7/2)*b^4-45*B*(a
*b)^(1/2)*x^(7/2)*a*b^3-63*A*(a*b)^(1/2)*x^(5/2)*a*b^3+63*B*(a*b)^(1/2)*x^(5/2)*
a^2*b^2+105*A*(a*b)^(1/2)*x^(3/2)*a^2*b^2-105*B*(a*b)^(1/2)*x^(3/2)*a^3*b-315*A*
(a*b)^(1/2)*x^(1/2)*a^3*b+315*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^4*b+315*B*(a*b)^
(1/2)*x^(1/2)*a^4-315*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)/((b*x+a)^2)^(1/2)/b^5
/(a*b)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291187, size = 1, normalized size = 0. \[ \left [-\frac{315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{315 \, b^{5}}, -\frac{2 \,{\left (315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/315*(315*(B*a^4 - A*a^3*b)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)
/(b*x + a)) - 2*(35*B*b^4*x^4 + 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x
^3 + 63*(B*a^2*b^2 - A*a*b^3)*x^2 - 105*(B*a^3*b - A*a^2*b^2)*x)*sqrt(x))/b^5, -
2/315*(315*(B*a^4 - A*a^3*b)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) - (35*B*b^4*x^4
+ 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 - A*a*b^3)
*x^2 - 105*(B*a^3*b - A*a^2*b^2)*x)*sqrt(x))/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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276741, size = 277, normalized size = 0.97 \[ -\frac{2 \,{\left (B a^{5}{\rm sign}\left (b x + a\right ) - A a^{4} b{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{2 \,{\left (35 \, B b^{8} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) - 45 \, B a b^{7} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + 45 \, A b^{8} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + 63 \, B a^{2} b^{6} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) - 63 \, A a b^{7} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) - 105 \, B a^{3} b^{5} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 105 \, A a^{2} b^{6} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 315 \, B a^{4} b^{4} \sqrt{x}{\rm sign}\left (b x + a\right ) - 315 \, A a^{3} b^{5} \sqrt{x}{\rm sign}\left (b x + a\right )\right )}}{315 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a^5*sign(b*x + a) - A*a^4*b*sign(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sq
rt(a*b)*b^5) + 2/315*(35*B*b^8*x^(9/2)*sign(b*x + a) - 45*B*a*b^7*x^(7/2)*sign(b
*x + a) + 45*A*b^8*x^(7/2)*sign(b*x + a) + 63*B*a^2*b^6*x^(5/2)*sign(b*x + a) -
63*A*a*b^7*x^(5/2)*sign(b*x + a) - 105*B*a^3*b^5*x^(3/2)*sign(b*x + a) + 105*A*a
^2*b^6*x^(3/2)*sign(b*x + a) + 315*B*a^4*b^4*sqrt(x)*sign(b*x + a) - 315*A*a^3*b
^5*sqrt(x)*sign(b*x + a))/b^9