3.309 \(\int \frac{(a+b x)^3 (A+B x)}{\sqrt{x}} \, dx\)

Optimal. Leaf size=83 \[ 2 a^3 A \sqrt{x}+\frac{2}{3} a^2 x^{3/2} (a B+3 A b)+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{9} b^3 B x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.103278, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ 2 a^3 A \sqrt{x}+\frac{2}{3} a^2 x^{3/2} (a B+3 A b)+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{9} b^3 B x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.5049, size = 82, normalized size = 0.99 \[ 2 A a^{3} \sqrt{x} + \frac{2 B b^{3} x^{\frac{9}{2}}}{9} + 2 a^{2} x^{\frac{3}{2}} \left (A b + \frac{B a}{3}\right ) + \frac{6 a b x^{\frac{5}{2}} \left (A b + B a\right )}{5} + \frac{2 b^{2} x^{\frac{7}{2}} \left (A b + 3 B a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0363696, size = 70, normalized size = 0.84 \[ \frac{2}{315} \sqrt{x} \left (105 a^3 (3 A+B x)+63 a^2 b x (5 A+3 B x)+27 a b^2 x^2 (7 A+5 B x)+5 b^3 x^3 (9 A+7 B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(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)))/315

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Maple [A]  time = 0.007, size = 76, normalized size = 0.9 \[{\frac{70\,B{b}^{3}{x}^{4}+90\,A{b}^{3}{x}^{3}+270\,B{x}^{3}a{b}^{2}+378\,aA{b}^{2}{x}^{2}+378\,B{x}^{2}{a}^{2}b+630\,{a}^{2}Abx+210\,{a}^{3}Bx+630\,{a}^{3}A}{315}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/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)

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Maxima [A]  time = 1.37516, size = 99, normalized size = 1.19 \[ \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + 2 \, A a^{3} \sqrt{x} + \frac{2}{7} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208043, size = 99, normalized size = 1.19 \[ \frac{2}{315} \,{\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/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 [A]  time = 10.3442, size = 110, normalized size = 1.33 \[ 2 A a^{3} \sqrt{x} + 2 A a^{2} b x^{\frac{3}{2}} + \frac{6 A a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 A b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B a^{3} x^{\frac{3}{2}}}{3} + \frac{6 B a^{2} b x^{\frac{5}{2}}}{5} + \frac{6 B a b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272453, size = 104, normalized size = 1.25 \[ \frac{2}{9} \, B b^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B a b^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B a^{2} b x^{\frac{5}{2}} + \frac{6}{5} \, A a b^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{2} b x^{\frac{3}{2}} + 2 \, A a^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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