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

Optimal. Leaf size=220 \[ \frac{2 b^2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{9 (a+b x)}+\frac{6 a b x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 (a+b x)}+\frac{2 b^3 B x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{2 a^3 A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

[Out]

(2*a^3*A*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a^2*(3*A*b +
a*B)*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (6*a*b*(A*b + a*B)*x
^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*x^(9/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*b^3*B*x^(11/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(11*(a + b*x))

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Rubi [A]  time = 0.24327, antiderivative size = 220, 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 \[ \frac{2 b^2 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{9 (a+b x)}+\frac{6 a b x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac{2 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 (a+b x)}+\frac{2 b^3 B x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{2 a^3 A x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a^2*(3*A*b +
a*B)*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (6*a*b*(A*b + a*B)*x
^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*x^(9/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*b^3*B*x^(11/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(11*(a + b*x))

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Rubi in Sympy [A]  time = 24.962, size = 223, normalized size = 1.01 \[ \frac{B x^{\frac{3}{2}} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{11 b} + \frac{32 a^{3} x^{\frac{3}{2}} \left (11 A b - 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3465 b \left (a + b x\right )} + \frac{16 a^{2} x^{\frac{3}{2}} \left (11 A b - 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{1155 b} + \frac{4 a x^{\frac{3}{2}} \left (3 a + 3 b x\right ) \left (11 A b - 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 b} + \frac{2 x^{\frac{3}{2}} \left (11 A b - 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**(3/2)*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(11*b) + 32*a**3*x*
*(3/2)*(11*A*b - 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(3465*b*(a + b*x)) + 16
*a**2*x**(3/2)*(11*A*b - 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(1155*b) + 4*a*
x**(3/2)*(3*a + 3*b*x)*(11*A*b - 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(693*b)
+ 2*x**(3/2)*(11*A*b - 3*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(99*b)

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Mathematica [A]  time = 0.0812325, size = 89, normalized size = 0.4 \[ \frac{2 x^{3/2} \sqrt{(a+b x)^2} \left (231 a^3 (5 A+3 B x)+297 a^2 b x (7 A+5 B x)+165 a b^2 x^2 (9 A+7 B x)+35 b^3 x^3 (11 A+9 B x)\right )}{3465 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*Sqrt[(a + b*x)^2]*(231*a^3*(5*A + 3*B*x) + 297*a^2*b*x*(7*A + 5*B*x)
+ 165*a*b^2*x^2*(9*A + 7*B*x) + 35*b^3*x^3*(11*A + 9*B*x)))/(3465*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[{\frac{630\,B{x}^{4}{b}^{3}+770\,A{b}^{3}{x}^{3}+2310\,B{x}^{3}a{b}^{2}+2970\,A{x}^{2}a{b}^{2}+2970\,B{x}^{2}{a}^{2}b+4158\,A{a}^{2}bx+1386\,{a}^{3}Bx+2310\,A{a}^{3}}{3465\, \left ( bx+a \right ) ^{3}}{x}^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*x^(3/2)*(315*B*b^3*x^4+385*A*b^3*x^3+1155*B*a*b^2*x^3+1485*A*a*b^2*x^2+14
85*B*a^2*b*x^2+2079*A*a^2*b*x+693*B*a^3*x+1155*A*a^3)*((b*x+a)^2)^(3/2)/(b*x+a)^
3

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Maxima [A]  time = 0.699898, size = 185, normalized size = 0.84 \[ \frac{2}{315} \,{\left (5 \,{\left (7 \, b^{3} x^{2} + 9 \, a b^{2} x\right )} x^{\frac{5}{2}} + 18 \,{\left (5 \, a b^{2} x^{2} + 7 \, a^{2} b x\right )} x^{\frac{3}{2}} + 21 \,{\left (3 \, a^{2} b x^{2} + 5 \, a^{3} x\right )} \sqrt{x}\right )} A + \frac{2}{3465} \,{\left (35 \,{\left (9 \, b^{3} x^{2} + 11 \, a b^{2} x\right )} x^{\frac{7}{2}} + 110 \,{\left (7 \, a b^{2} x^{2} + 9 \, a^{2} b x\right )} x^{\frac{5}{2}} + 99 \,{\left (5 \, a^{2} b x^{2} + 7 \, a^{3} x\right )} x^{\frac{3}{2}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(5*(7*b^3*x^2 + 9*a*b^2*x)*x^(5/2) + 18*(5*a*b^2*x^2 + 7*a^2*b*x)*x^(3/2)
+ 21*(3*a^2*b*x^2 + 5*a^3*x)*sqrt(x))*A + 2/3465*(35*(9*b^3*x^2 + 11*a*b^2*x)*x^
(7/2) + 110*(7*a*b^2*x^2 + 9*a^2*b*x)*x^(5/2) + 99*(5*a^2*b*x^2 + 7*a^3*x)*x^(3/
2))*B

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Fricas [A]  time = 0.303748, size = 103, normalized size = 0.47 \[ \frac{2}{3465} \,{\left (315 \, B b^{3} x^{5} + 1155 \, A a^{3} x + 385 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 1485 \,{\left (B a^{2} b + A a b^{2}\right )} x^{3} + 693 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*B*b^3*x^5 + 1155*A*a^3*x + 385*(3*B*a*b^2 + A*b^3)*x^4 + 1485*(B*a^2
*b + A*a*b^2)*x^3 + 693*(B*a^3 + 3*A*a^2*b)*x^2)*sqrt(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)*(A + B*x)*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.272203, size = 169, normalized size = 0.77 \[ \frac{2}{11} \, B b^{3} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, B a b^{2} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{7} \, B a^{2} b x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{7} \, A a b^{2} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{5} \, A a^{2} b x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*B*b^3*x^(11/2)*sign(b*x + a) + 2/3*B*a*b^2*x^(9/2)*sign(b*x + a) + 2/9*A*b^
3*x^(9/2)*sign(b*x + a) + 6/7*B*a^2*b*x^(7/2)*sign(b*x + a) + 6/7*A*a*b^2*x^(7/2
)*sign(b*x + a) + 2/5*B*a^3*x^(5/2)*sign(b*x + a) + 6/5*A*a^2*b*x^(5/2)*sign(b*x
+ a) + 2/3*A*a^3*x^(3/2)*sign(b*x + a)