3.331 \(\int \frac {(a+b x)^2 (A+B x)}{\sqrt {x}} \, dx\)

Optimal. Leaf size=61 \[ 2 a^2 A \sqrt {x}+\frac {2}{5} b x^{5/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{7} b^2 B x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {76} \[ 2 a^2 A \sqrt {x}+\frac {2}{5} b x^{5/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{7} b^2 B x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 76

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

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Mathematica [A]  time = 0.02, size = 51, normalized size = 0.84 \[ \frac {2}{105} \sqrt {x} \left (35 a^2 (3 A+B x)+14 a b x (5 A+3 B x)+3 b^2 x^2 (7 A+5 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.93, size = 51, normalized size = 0.84 \[ \frac {2}{105} \, {\left (15 \, B b^{2} x^{3} + 105 \, A a^{2} + 21 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 35 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.22, size = 53, normalized size = 0.87 \[ \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{5} \, B a b x^{\frac {5}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a b x^{\frac {3}{2}} + 2 \, A a^{2} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^(1/2),x, algorithm="giac")

[Out]

2/7*B*b^2*x^(7/2) + 4/5*B*a*b*x^(5/2) + 2/5*A*b^2*x^(5/2) + 2/3*B*a^2*x^(3/2) + 4/3*A*a*b*x^(3/2) + 2*A*a^2*sq
rt(x)

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maple [A]  time = 0.01, size = 52, normalized size = 0.85 \[ \frac {2 \left (15 B \,b^{2} x^{3}+21 A \,b^{2} x^{2}+42 B a b \,x^{2}+70 A a b x +35 B \,a^{2} x +105 a^{2} A \right ) \sqrt {x}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*x^(1/2)*(15*B*b^2*x^3+21*A*b^2*x^2+42*B*a*b*x^2+70*A*a*b*x+35*B*a^2*x+105*A*a^2)

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maxima [A]  time = 0.80, size = 51, normalized size = 0.84 \[ \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + 2 \, A a^{2} \sqrt {x} + \frac {2}{5} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 51, normalized size = 0.84 \[ x^{3/2}\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )+x^{5/2}\,\left (\frac {2\,A\,b^2}{5}+\frac {4\,B\,a\,b}{5}\right )+2\,A\,a^2\,\sqrt {x}+\frac {2\,B\,b^2\,x^{7/2}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.55, size = 78, normalized size = 1.28 \[ 2 A a^{2} \sqrt {x} + \frac {4 A a b x^{\frac {3}{2}}}{3} + \frac {2 A b^{2} x^{\frac {5}{2}}}{5} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a b x^{\frac {5}{2}}}{5} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(3/2)/3 + 2*A*b**2*x**(5/2)/5 + 2*B*a**2*x**(3/2)/3 + 4*B*a*b*x**(5/2)/5 + 2*B*b
**2*x**(7/2)/7

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