3.4.90 \(\int x \sqrt {a+b x} (A+B x) \, dx\) [390]

Optimal. Leaf size=67 \[ -\frac {2 a (A b-a B) (a+b x)^{3/2}}{3 b^3}+\frac {2 (A b-2 a B) (a+b x)^{5/2}}{5 b^3}+\frac {2 B (a+b x)^{7/2}}{7 b^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \begin {gather*} \frac {2 (a+b x)^{5/2} (A b-2 a B)}{5 b^3}-\frac {2 a (a+b x)^{3/2} (A b-a B)}{3 b^3}+\frac {2 B (a+b x)^{7/2}}{7 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + b*x]*(A + B*x),x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 49, normalized size = 0.73 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (8 a^2 B+3 b^2 x (7 A+5 B x)-2 a b (7 A+6 B x)\right )}{105 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(3/2)*(8*a^2*B + 3*b^2*x*(7*A + 5*B*x) - 2*a*b*(7*A + 6*B*x)))/(105*b^3)

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Maple [A]
time = 0.06, size = 52, normalized size = 0.78

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-15 b^{2} B \,x^{2}-21 b^{2} A x +12 a b B x +14 a b A -8 a^{2} B \right )}{105 b^{3}}\)	\(47\)
derivativedivides	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\)	\(52\)
default	\(\frac {\frac {2 B \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a \left (A b -B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\)	\(52\)
trager	\(-\frac {2 \left (-15 b^{3} B \,x^{3}-21 A \,b^{3} x^{2}-3 B a \,b^{2} x^{2}-7 a \,b^{2} A x +4 a^{2} b B x +14 a^{2} b A -8 a^{3} B \right ) \sqrt {b x +a}}{105 b^{3}}\)	\(71\)
risch	\(-\frac {2 \left (-15 b^{3} B \,x^{3}-21 A \,b^{3} x^{2}-3 B a \,b^{2} x^{2}-7 a \,b^{2} A x +4 a^{2} b B x +14 a^{2} b A -8 a^{3} B \right ) \sqrt {b x +a}}{105 b^{3}}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 54, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} B - 21 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 35 \, {\left (B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{105 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(b*x + a)^(7/2)*B - 21*(2*B*a - A*b)*(b*x + a)^(5/2) + 35*(B*a^2 - A*a*b)*(b*x + a)^(3/2))/b^3

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Fricas [A]
time = 0.81, size = 71, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (15 \, B b^{3} x^{3} + 8 \, B a^{3} - 14 \, A a^{2} b + 3 \, {\left (B a b^{2} + 7 \, A b^{3}\right )} x^{2} - {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x\right )} \sqrt {b x + a}}{105 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*B*b^3*x^3 + 8*B*a^3 - 14*A*a^2*b + 3*(B*a*b^2 + 7*A*b^3)*x^2 - (4*B*a^2*b - 7*A*a*b^2)*x)*sqrt(b*x +
a)/b^3

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Sympy [A]
time = 1.19, size = 63, normalized size = 0.94 \begin {gather*} \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {7}{2}}}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (A b - 2 B a\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- A a b + B a^{2}\right )}{3 b}\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (56) = 112\).
time = 2.25, size = 158, normalized size = 2.36 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A a}{b} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B a}{b^{2}} + \frac {7 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} A}{b} + \frac {3 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} B}{b^{2}}\right )}}{105 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(35*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*A*a/b + 7*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(
b*x + a)*a^2)*B*a/b^2 + 7*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*A/b + 3*(5*(b*x +
a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*B/b^2)/b

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Mupad [B]
time = 0.07, size = 52, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{3/2}\,\left (35\,B\,a^2+15\,B\,{\left (a+b\,x\right )}^2-35\,A\,a\,b+21\,A\,b\,\left (a+b\,x\right )-42\,B\,a\,\left (a+b\,x\right )\right )}{105\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(A + B*x)*(a + b*x)^(1/2),x)

[Out]

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

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