3.18.29 \(\int (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx\) [1729]

Optimal. Leaf size=128 \[ -\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{3/2}}{3 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{5/2}}{5 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{7/2}}{7 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4} \]

[Out]

-2/3*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(3/2)/e^4+2/5*(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(5/2)/e^4-2/7*b
*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(7/2)/e^4+2/9*b^2*B*(e*x+d)^(9/2)/e^4

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Rubi [A]
time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac {2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(5/2))/(5*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(7/2))/(7*e^4) + (2*b^2*B*(d + e*x)^(9/2))/(9*e^4)

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 (a+b x)^2 (A+B x) \sqrt {d+e x} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) \sqrt {d+e x}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{3/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{5/2}}{e^3}+\frac {b^2 B (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{3/2}}{3 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{5/2}}{5 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{7/2}}{7 e^4}+\frac {2 b^2 B (d+e x)^{9/2}}{9 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 138, normalized size = 1.08 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (21 a^2 e^2 (-2 B d+5 A e+3 B e x)+6 a b e \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+b^2 \left (3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(21*a^2*e^2*(-2*B*d + 5*A*e + 3*B*e*x) + 6*a*b*e*(7*A*e*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*
x + 15*e^2*x^2)) + b^2*(3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^
3*x^3))))/(315*e^4)

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Maple [A]
time = 0.09, size = 122, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\)	\(122\)
default	\(\frac {\frac {2 b^{2} B \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a e -b d \right ) b B +b^{2} \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right )^{2} B +2 \left (a e -b d \right ) b \left (A e -B d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a e -b d \right )^{2} \left (A e -B d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\)	\(122\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 b^{2} B \,x^{3} e^{3}+45 A \,b^{2} e^{3} x^{2}+90 B a b \,e^{3} x^{2}-30 B \,b^{2} d \,e^{2} x^{2}+126 A a b \,e^{3} x -36 A \,b^{2} d \,e^{2} x +63 B \,a^{2} e^{3} x -72 B a b d \,e^{2} x +24 B \,b^{2} d^{2} e x +105 a^{2} A \,e^{3}-84 A a b d \,e^{2}+24 A \,b^{2} d^{2} e -42 B \,a^{2} d \,e^{2}+48 B a b \,d^{2} e -16 b^{2} B \,d^{3}\right )}{315 e^{4}}\)	\(169\)
trager	\(\frac {2 \left (35 b^{2} B \,e^{4} x^{4}+45 A \,b^{2} e^{4} x^{3}+90 B a b \,e^{4} x^{3}+5 b^{2} B d \,e^{3} x^{3}+126 A a b \,e^{4} x^{2}+9 A \,b^{2} d \,e^{3} x^{2}+63 B \,a^{2} e^{4} x^{2}+18 B a b d \,e^{3} x^{2}-6 B \,b^{2} d^{2} e^{2} x^{2}+105 a^{2} A \,e^{4} x +42 A a b d \,e^{3} x -12 A \,b^{2} d^{2} e^{2} x +21 B \,a^{2} d \,e^{3} x -24 B a b \,d^{2} e^{2} x +8 b^{2} B \,d^{3} e x +105 a^{2} A d \,e^{3}-84 A a b \,d^{2} e^{2}+24 A \,b^{2} d^{3} e -42 B \,a^{2} d^{2} e^{2}+48 B a b \,d^{3} e -16 b^{2} B \,d^{4}\right ) \sqrt {e x +d}}{315 e^{4}}\)	\(253\)
risch	\(\frac {2 \left (35 b^{2} B \,e^{4} x^{4}+45 A \,b^{2} e^{4} x^{3}+90 B a b \,e^{4} x^{3}+5 b^{2} B d \,e^{3} x^{3}+126 A a b \,e^{4} x^{2}+9 A \,b^{2} d \,e^{3} x^{2}+63 B \,a^{2} e^{4} x^{2}+18 B a b d \,e^{3} x^{2}-6 B \,b^{2} d^{2} e^{2} x^{2}+105 a^{2} A \,e^{4} x +42 A a b d \,e^{3} x -12 A \,b^{2} d^{2} e^{2} x +21 B \,a^{2} d \,e^{3} x -24 B a b \,d^{2} e^{2} x +8 b^{2} B \,d^{3} e x +105 a^{2} A d \,e^{3}-84 A a b \,d^{2} e^{2}+24 A \,b^{2} d^{3} e -42 B \,a^{2} d^{2} e^{2}+48 B a b \,d^{3} e -16 b^{2} B \,d^{4}\right ) \sqrt {e x +d}}{315 e^{4}}\)	\(253\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.42, size = 168, normalized size = 1.31 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{2} - 45 \, {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 63 \, {\left (3 \, B b^{2} d^{2} + B a^{2} e^{2} + 2 \, A a b e^{2} - 2 \, {\left (2 \, B a b e + A b^{2} e\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 105 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b e + A b^{2} e\right )} d^{2} + {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(x*e + d)^(9/2)*B*b^2 - 45*(3*B*b^2*d - 2*B*a*b*e - A*b^2*e)*(x*e + d)^(7/2) + 63*(3*B*b^2*d^2 + B*a
^2*e^2 + 2*A*a*b*e^2 - 2*(2*B*a*b*e + A*b^2*e)*d)*(x*e + d)^(5/2) - 105*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b*e +
A*b^2*e)*d^2 + (B*a^2*e^2 + 2*A*a*b*e^2)*d)*(x*e + d)^(3/2))*e^(-4)

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Fricas [A]
time = 1.29, size = 208, normalized size = 1.62 \begin {gather*} -\frac {2}{315} \, {\left (16 \, B b^{2} d^{4} - {\left (35 \, B b^{2} x^{4} + 105 \, A a^{2} x + 45 \, {\left (2 \, B a b + A b^{2}\right )} x^{3} + 63 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} e^{4} - {\left (5 \, B b^{2} d x^{3} + 105 \, A a^{2} d + 9 \, {\left (2 \, B a b + A b^{2}\right )} d x^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} d x\right )} e^{3} + 6 \, {\left (B b^{2} d^{2} x^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} x + 7 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} e^{2} - 8 \, {\left (B b^{2} d^{3} x + 3 \, {\left (2 \, B a b + A b^{2}\right )} d^{3}\right )} e\right )} \sqrt {x e + d} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(16*B*b^2*d^4 - (35*B*b^2*x^4 + 105*A*a^2*x + 45*(2*B*a*b + A*b^2)*x^3 + 63*(B*a^2 + 2*A*a*b)*x^2)*e^4
- (5*B*b^2*d*x^3 + 105*A*a^2*d + 9*(2*B*a*b + A*b^2)*d*x^2 + 21*(B*a^2 + 2*A*a*b)*d*x)*e^3 + 6*(B*b^2*d^2*x^2
+ 2*(2*B*a*b + A*b^2)*d^2*x + 7*(B*a^2 + 2*A*a*b)*d^2)*e^2 - 8*(B*b^2*d^3*x + 3*(2*B*a*b + A*b^2)*d^3)*e)*sqrt
(x*e + d)*e^(-4)

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Sympy [A]
time = 2.23, size = 201, normalized size = 1.57 \begin {gather*} \frac {2 \left (\frac {B b^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{3 e^{3}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(B*b**2*(d + e*x)**(9/2)/(9*e**3) + (d + e*x)**(7/2)*(A*b**2*e + 2*B*a*b*e - 3*B*b**2*d)/(7*e**3) + (d + e*x
)**(5/2)*(2*A*a*b*e**2 - 2*A*b**2*d*e + B*a**2*e**2 - 4*B*a*b*d*e + 3*B*b**2*d**2)/(5*e**3) + (d + e*x)**(3/2)
*(A*a**2*e**3 - 2*A*a*b*d*e**2 + A*b**2*d**2*e - B*a**2*d*e**2 + 2*B*a*b*d**2*e - B*b**2*d**3)/(3*e**3))/e

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (119) = 238\).
time = 0.83, size = 511, normalized size = 3.99 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{2} d e^{\left (-1\right )} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a b d e^{\left (-1\right )} + 42 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a b d e^{\left (-2\right )} + 21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A b^{2} d e^{\left (-2\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B b^{2} d e^{\left (-3\right )} + 21 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{2} e^{\left (-1\right )} + 42 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a b e^{\left (-1\right )} + 18 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a b e^{\left (-2\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A b^{2} e^{\left (-2\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B b^{2} e^{\left (-3\right )} + 315 \, \sqrt {x e + d} A a^{2} d + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^2*d*e^(-1) + 210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*
a*b*d*e^(-1) + 42*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*b*d*e^(-2) + 21*(3*(x*
e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d*e^(-2) + 9*(5*(x*e + d)^(7/2) - 21*(x*e +
d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*b^2*d*e^(-3) + 21*(3*(x*e + d)^(5/2) - 10*(x*e +
d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*e^(-1) + 42*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e
+ d)*d^2)*A*a*b*e^(-1) + 18*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e +
d)*d^3)*B*a*b*e^(-2) + 9*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d
)*d^3)*A*b^2*e^(-2) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3
/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b^2*e^(-3) + 315*sqrt(x*e + d)*A*a^2*d + 105*((x*e + d)^(3/2) - 3*sqrt(x*e
+ d)*d)*A*a^2)*e^(-1)

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Mupad [B]
time = 0.07, size = 115, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{7\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{5\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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