3.784 \(\int x^{3/2} (A+B x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=120 \[ \frac{2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 a A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]

[Out]

(2*a*A*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*(A*b + a*B)*x^(
7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b*B*x^(9/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(9*(a + b*x))

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Rubi [A]  time = 0.154175, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 a A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*A*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*(A*b + a*B)*x^(
7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b*B*x^(9/2)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(9*(a + b*x))

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Rubi in Sympy [A]  time = 18.9906, size = 126, normalized size = 1.05 \[ \frac{B x^{\frac{5}{2}} \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9 b} + \frac{4 a x^{\frac{5}{2}} \left (9 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 b \left (a + b x\right )} + \frac{2 x^{\frac{5}{2}} \left (9 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)*((b*x+a)**2)**(1/2),x)

[Out]

B*x**(5/2)*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9*b) + 4*a*x**(5/2)*(
9*A*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*b*(a + b*x)) + 2*x**(5/2)*(
9*A*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*b)

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Mathematica [A]  time = 0.0424896, size = 51, normalized size = 0.42 \[ \frac{2 x^{5/2} \sqrt{(a+b x)^2} (9 a (7 A+5 B x)+5 b x (9 A+7 B x))}{315 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(5/2)*Sqrt[(a + b*x)^2]*(9*a*(7*A + 5*B*x) + 5*b*x*(9*A + 7*B*x)))/(315*(a
+ b*x))

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Maple [A]  time = 0.006, size = 44, normalized size = 0.4 \[{\frac{70\,Bb{x}^{2}+90\,Abx+90\,aBx+126\,aA}{315\,bx+315\,a}{x}^{{\frac{5}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)*((b*x+a)^2)^(1/2),x)

[Out]

2/315*x^(5/2)*(35*B*b*x^2+45*A*b*x+45*B*a*x+63*A*a)*((b*x+a)^2)^(1/2)/(b*x+a)

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Maxima [A]  time = 0.695614, size = 47, normalized size = 0.39 \[ \frac{2}{63} \,{\left (7 \, b x^{2} + 9 \, a x\right )} B x^{\frac{5}{2}} + \frac{2}{35} \,{\left (5 \, b x^{2} + 7 \, a x\right )} A x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*b*x^2 + 9*a*x)*B*x^(5/2) + 2/35*(5*b*x^2 + 7*a*x)*A*x^(3/2)

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Fricas [A]  time = 0.30707, size = 43, normalized size = 0.36 \[ \frac{2}{315} \,{\left (35 \, B b x^{4} + 63 \, A a x^{2} + 45 \,{\left (B a + A b\right )} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*b*x^4 + 63*A*a*x^2 + 45*(B*a + A*b)*x^3)*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(x**(3/2)*(B*x+A)*((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283577, size = 72, normalized size = 0.6 \[ \frac{2}{9} \, B b x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, B a x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, A b x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{5} \, A a x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*B*b*x^(9/2)*sign(b*x + a) + 2/7*B*a*x^(7/2)*sign(b*x + a) + 2/7*A*b*x^(7/2)*
sign(b*x + a) + 2/5*A*a*x^(5/2)*sign(b*x + a)