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

Optimal. Leaf size=53 \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} x^{5/2} (A c+b B)+\frac{2}{7} B c x^{7/2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0634002, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} x^{5/2} (A c+b B)+\frac{2}{7} B c x^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.39045, size = 58, normalized size = 1.09 \[ 2 A a \sqrt{x} + \frac{2 B c x^{\frac{7}{2}}}{7} + x^{\frac{5}{2}} \left (\frac{2 A c}{5} + \frac{2 B b}{5}\right ) + x^{\frac{3}{2}} \left (\frac{2 A b}{3} + \frac{2 B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0300634, size = 46, normalized size = 0.87 \[ \frac{2}{105} \sqrt{x} (35 a (3 A+B x)+x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(35*a*(3*A + B*x) + x*(7*A*(5*b + 3*c*x) + 3*B*x*(7*b + 5*c*x))))/105

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 42, normalized size = 0.8 \[{\frac{30\,Bc{x}^{3}+42\,Ac{x}^{2}+42\,Bb{x}^{2}+70\,Abx+70\,aBx+210\,aA}{105}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*x^(1/2)*(15*B*c*x^3+21*A*c*x^2+21*B*b*x^2+35*A*b*x+35*B*a*x+105*A*a)

_______________________________________________________________________________________

Maxima [A]  time = 0.708306, size = 53, normalized size = 1. \[ \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b + A c\right )} x^{\frac{5}{2}} + 2 \, A a \sqrt{x} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.289893, size = 53, normalized size = 1. \[ \frac{2}{105} \,{\left (15 \, B c x^{3} + 21 \,{\left (B b + A c\right )} x^{2} + 105 \, A a + 35 \,{\left (B a + A b\right )} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*B*c*x^3 + 21*(B*b + A*c)*x^2 + 105*A*a + 35*(B*a + A*b)*x)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 4.21101, size = 68, normalized size = 1.28 \[ 2 A a \sqrt{x} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 A c x^{\frac{5}{2}}}{5} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{5}{2}}}{5} + \frac{2 B c x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268284, size = 58, normalized size = 1.09 \[ \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \, B b x^{\frac{5}{2}} + \frac{2}{5} \, A c x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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