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

Optimal. Leaf size=218 \[ \frac{2 b^2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac{6 a b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{2 a^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{3 (a+b x)}+\frac{2 b^3 B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{2 a^3 A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

(2*a^3*A*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*a^2*(3*A*b + a*B)
*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (6*a*b*(A*b + a*B)*x^(5/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*x^(7/2)*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b^3*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 = 0.243373, antiderivative size = 218, 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 \[ \frac{2 b^2 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 (a+b x)}+\frac{6 a b x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac{2 a^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{3 (a+b x)}+\frac{2 b^3 B x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{2 a^3 A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*a^2*(3*A*b + a*B)
*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (6*a*b*(A*b + a*B)*x^(5/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*b^2*(A*b + 3*a*B)*x^(7/2)*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*b^3*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 = 24.9063, size = 216, normalized size = 0.99 \[ \frac{B \sqrt{x} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{9 b} + \frac{32 a^{3} \sqrt{x} \left (9 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 b \left (a + b x\right )} + \frac{16 a^{2} \sqrt{x} \left (9 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 b} + \frac{4 a \sqrt{x} \left (3 a + 3 b x\right ) \left (9 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{315 b} + \frac{2 \sqrt{x} \left (9 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63 b} \]

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*sqrt(x)*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(9*b) + 32*a**3*sqrt
(x)*(9*A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*b*(a + b*x)) + 16*a**2*s
qrt(x)*(9*A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*b) + 4*a*sqrt(x)*(3*a
+ 3*b*x)*(9*A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(315*b) + 2*sqrt(x)*(9*
A*b - B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(63*b)

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Mathematica [A]  time = 0.0819355, size = 88, normalized size = 0.4 \[ \frac{2 \sqrt{x} \sqrt{(a+b x)^2} \left (105 a^3 (3 A+B x)+63 a^2 b x (5 A+3 B x)+27 a b^2 x^2 (7 A+5 B x)+5 b^3 x^3 (9 A+7 B x)\right )}{315 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*Sqrt[(a + b*x)^2]*(105*a^3*(3*A + B*x) + 63*a^2*b*x*(5*A + 3*B*x) + 2
7*a*b^2*x^2*(7*A + 5*B*x) + 5*b^3*x^3*(9*A + 7*B*x)))/(315*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[{\frac{70\,B{x}^{4}{b}^{3}+90\,A{b}^{3}{x}^{3}+270\,B{x}^{3}a{b}^{2}+378\,A{x}^{2}a{b}^{2}+378\,B{x}^{2}{a}^{2}b+630\,A{a}^{2}bx+210\,{a}^{3}Bx+630\,A{a}^{3}}{315\, \left ( bx+a \right ) ^{3}}\sqrt{x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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]

2/315*x^(1/2)*(35*B*b^3*x^4+45*A*b^3*x^3+135*B*a*b^2*x^3+189*A*a*b^2*x^2+189*B*a
^2*b*x^2+315*A*a^2*b*x+105*B*a^3*x+315*A*a^3)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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Maxima [A]  time = 0.702544, size = 184, normalized size = 0.84 \[ \frac{2}{105} \,{\left (3 \,{\left (5 \, b^{3} x^{2} + 7 \, a b^{2} x\right )} x^{\frac{3}{2}} + 14 \,{\left (3 \, a b^{2} x^{2} + 5 \, a^{2} b x\right )} \sqrt{x} + \frac{35 \,{\left (a^{2} b x^{2} + 3 \, a^{3} x\right )}}{\sqrt{x}}\right )} A + \frac{2}{315} \,{\left (5 \,{\left (7 \, b^{3} x^{2} + 9 \, a b^{2} x\right )} x^{\frac{5}{2}} + 18 \,{\left (5 \, a b^{2} x^{2} + 7 \, a^{2} b x\right )} x^{\frac{3}{2}} + 21 \,{\left (3 \, a^{2} b x^{2} + 5 \, a^{3} x\right )} \sqrt{x}\right )} B \]

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]

2/105*(3*(5*b^3*x^2 + 7*a*b^2*x)*x^(3/2) + 14*(3*a*b^2*x^2 + 5*a^2*b*x)*sqrt(x)
+ 35*(a^2*b*x^2 + 3*a^3*x)/sqrt(x))*A + 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))*B

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Fricas [A]  time = 0.307745, size = 99, normalized size = 0.45 \[ \frac{2}{315} \,{\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} \sqrt{x} \]

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]

2/315*(35*B*b^3*x^4 + 315*A*a^3 + 45*(3*B*a*b^2 + A*b^3)*x^3 + 189*(B*a^2*b + A*
a*b^2)*x^2 + 105*(B*a^3 + 3*A*a^2*b)*x)*sqrt(x)

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

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

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GIAC/XCAS [A]  time = 0.272508, size = 169, normalized size = 0.78 \[ \frac{2}{9} \, B b^{3} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{7} \, B a b^{2} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{5} \, B a^{2} b x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{6}{5} \, A a b^{2} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 2 \, A a^{2} b x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 2 \, A a^{3} \sqrt{x}{\rm sign}\left (b x + a\right ) \]

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]

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