\(\int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx\) [339]

Optimal result

Integrand size = 18, antiderivative size = 83 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=2 a^3 A \sqrt {x}+\frac {2}{3} a^2 (3 A b+a B) x^{3/2}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{7} b^2 (A b+3 a B) x^{7/2}+\frac {2}{9} b^3 B x^{9/2} \]

[Out]

2/3*a^2*(3*A*b+B*a)*x^(3/2)+6/5*a*b*(A*b+B*a)*x^(5/2)+2/7*b^2*(A*b+3*B*a)*x^(7/2)+2/9*b^3*B*x^(9/2)+2*a^3*A*x^
(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=2 a^3 A \sqrt {x}+\frac {2}{3} a^2 x^{3/2} (a B+3 A b)+\frac {2}{7} b^2 x^{7/2} (3 a B+A b)+\frac {6}{5} a b x^{5/2} (a B+A b)+\frac {2}{9} b^3 B x^{9/2} \]

[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

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 A}{\sqrt {x}}+a^2 (3 A b+a B) \sqrt {x}+3 a b (A b+a B) x^{3/2}+b^2 (A b+3 a B) x^{5/2}+b^3 B x^{7/2}\right ) \, dx \\ & = 2 a^3 A \sqrt {x}+\frac {2}{3} a^2 (3 A b+a B) x^{3/2}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{7} b^2 (A b+3 a B) x^{7/2}+\frac {2}{9} b^3 B x^{9/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=\frac {2}{315} \sqrt {x} \left (105 a^3 (3 A+B x)+63 a^2 b x (5 A+3 B x)+27 a b^2 x^2 (7 A+5 B x)+5 b^3 x^3 (9 A+7 B x)\right ) \]

[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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90

method	result	size
trager	\(\left (\frac {2}{9} b^{3} B \,x^{4}+\frac {2}{7} A \,b^{3} x^{3}+\frac {6}{7} B a \,b^{2} x^{3}+\frac {6}{5} a A \,b^{2} x^{2}+\frac {6}{5} B \,a^{2} b \,x^{2}+2 a^{2} A b x +\frac {2}{3} a^{3} B x +2 a^{3} A \right ) \sqrt {x}\)	\(75\)
gosper	\(\frac {2 \sqrt {x}\, \left (35 b^{3} B \,x^{4}+45 A \,b^{3} x^{3}+135 B a \,b^{2} x^{3}+189 a A \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+315 a^{2} A b x +105 a^{3} B x +315 a^{3} A \right )}{315}\)	\(76\)
derivativedivides	\(\frac {2 b^{3} B \,x^{\frac {9}{2}}}{9}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {3}{2}}}{3}+2 a^{3} A \sqrt {x}\)	\(76\)
default	\(\frac {2 b^{3} B \,x^{\frac {9}{2}}}{9}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {3}{2}}}{3}+2 a^{3} A \sqrt {x}\)	\(76\)
risch	\(\frac {2 \sqrt {x}\, \left (35 b^{3} B \,x^{4}+45 A \,b^{3} x^{3}+135 B a \,b^{2} x^{3}+189 a A \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+315 a^{2} A b x +105 a^{3} B x +315 a^{3} A \right )}{315}\)	\(76\)

[In]

int((b*x+a)^3*(B*x+A)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

(2/9*b^3*B*x^4+2/7*A*b^3*x^3+6/7*B*a*b^2*x^3+6/5*a*A*b^2*x^2+6/5*B*a^2*b*x^2+2*a^2*A*b*x+2/3*a^3*B*x+2*a^3*A)*
x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=\frac {2}{315} \, {\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} \sqrt {x} \]

[In]

integrate((b*x+a)^3*(B*x+A)/x^(1/2),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)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=2 A a^{3} \sqrt {x} + 2 A a^{2} b x^{\frac {3}{2}} + \frac {6 A a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 A b^{3} x^{\frac {7}{2}}}{7} + \frac {2 B a^{3} x^{\frac {3}{2}}}{3} + \frac {6 B a^{2} b x^{\frac {5}{2}}}{5} + \frac {6 B a b^{2} x^{\frac {7}{2}}}{7} + \frac {2 B b^{3} x^{\frac {9}{2}}}{9} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=\frac {2}{9} \, B b^{3} x^{\frac {9}{2}} + 2 \, A a^{3} \sqrt {x} + \frac {2}{7} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {7}{2}} + \frac {6}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {3}{2}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/x^(1/2),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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=\frac {2}{9} \, B b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, B a b^{2} x^{\frac {7}{2}} + \frac {2}{7} \, A b^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B a^{2} b x^{\frac {5}{2}} + \frac {6}{5} \, A a b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}} + 2 \, A a^{3} \sqrt {x} \]

[In]

integrate((b*x+a)^3*(B*x+A)/x^(1/2),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)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx=x^{3/2}\,\left (\frac {2\,B\,a^3}{3}+2\,A\,b\,a^2\right )+x^{7/2}\,\left (\frac {2\,A\,b^3}{7}+\frac {6\,B\,a\,b^2}{7}\right )+2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,b^3\,x^{9/2}}{9}+\frac {6\,a\,b\,x^{5/2}\,\left (A\,b+B\,a\right )}{5} \]

[In]

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

[Out]

x^(3/2)*((2*B*a^3)/3 + 2*A*a^2*b) + x^(7/2)*((2*A*b^3)/7 + (6*B*a*b^2)/7) + 2*A*a^3*x^(1/2) + (2*B*b^3*x^(9/2)
)/9 + (6*a*b*x^(5/2)*(A*b + B*a))/5