3.11.71 \(\int (A+B x) \sqrt {d+e x} (b x+c x^2) \, dx\)

Optimal. Leaf size=126 \[ -\frac {2 (d+e x)^{7/2} (-A c e-b B e+3 B c d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {2 d (d+e x)^{3/2} (B d-A e) (c d-b e)}{3 e^4}+\frac {2 B c (d+e x)^{9/2}}{9 e^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{7/2} (-A c e-b B e+3 B c d)}{7 e^4}+\frac {2 (d+e x)^{5/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {2 d (d+e x)^{3/2} (B d-A e) (c d-b e)}{3 e^4}+\frac {2 B c (d+e x)^{9/2}}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 113, normalized size = 0.90 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (3 A e \left (7 b e (3 e x-2 d)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+B \left (3 b e \left (8 d^2-12 d e x+15 e^2 x^2\right )+c \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)*Sqrt[d + e*x]*(b*x + c*x^2),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.08, size = 141, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (63 A b e^2 (d+e x)-105 A b d e^2+105 A c d^2 e-126 A c d e (d+e x)+45 A c e (d+e x)^2+105 b B d^2 e-126 b B d e (d+e x)+45 b B e (d+e x)^2-105 B c d^3+189 B c d^2 (d+e x)-135 B c d (d+e x)^2+35 B c (d+e x)^3\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-105*B*c*d^3 + 105*b*B*d^2*e + 105*A*c*d^2*e - 105*A*b*d*e^2 + 189*B*c*d^2*(d + e*x) - 126
*b*B*d*e*(d + e*x) - 126*A*c*d*e*(d + e*x) + 63*A*b*e^2*(d + e*x) - 135*B*c*d*(d + e*x)^2 + 45*b*B*e*(d + e*x)
^2 + 45*A*c*e*(d + e*x)^2 + 35*B*c*(d + e*x)^3))/(315*e^4)

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 148, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left (35 \, B c e^{4} x^{4} - 16 \, B c d^{4} - 42 \, A b d^{2} e^{2} + 24 \, {\left (B b + A c\right )} d^{3} e + 5 \, {\left (B c d e^{3} + 9 \, {\left (B b + A c\right )} e^{4}\right )} x^{3} - 3 \, {\left (2 \, B c d^{2} e^{2} - 21 \, A b e^{4} - 3 \, {\left (B b + A c\right )} d e^{3}\right )} x^{2} + {\left (8 \, B c d^{3} e + 21 \, A b d e^{3} - 12 \, {\left (B b + A c\right )} d^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*c*e^4*x^4 - 16*B*c*d^4 - 42*A*b*d^2*e^2 + 24*(B*b + A*c)*d^3*e + 5*(B*c*d*e^3 + 9*(B*b + A*c)*e^4)
*x^3 - 3*(2*B*c*d^2*e^2 - 21*A*b*e^4 - 3*(B*b + A*c)*d*e^3)*x^2 + (8*B*c*d^3*e + 21*A*b*d*e^3 - 12*(B*b + A*c)
*d^2*e^2)*x)*sqrt(e*x + d)/e^4

________________________________________________________________________________________

giac [B]  time = 0.17, size = 386, normalized size = 3.06 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A b d e^{\left (-1\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 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 c 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 c 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 )} A b e^{\left (-1\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 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 c 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 c e^{\left (-3\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*d*e^(-1) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*B*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*c
*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*c*d
*e^(-3) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b*e^(-1) + 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*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*c*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*c*e^(-3))*e^(
-1)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 121, normalized size = 0.96 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (-35 B c \,x^{3} e^{3}-45 A c \,e^{3} x^{2}-45 B b \,e^{3} x^{2}+30 B c d \,e^{2} x^{2}-63 A b \,e^{3} x +36 A c d \,e^{2} x +36 B b d \,e^{2} x -24 B c \,d^{2} e x +42 A b d \,e^{2}-24 A c \,d^{2} e -24 B b \,d^{2} e +16 B c \,d^{3}\right )}{315 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(e*x+d)^(3/2)*(-35*B*c*e^3*x^3-45*A*c*e^3*x^2-45*B*b*e^3*x^2+30*B*c*d*e^2*x^2-63*A*b*e^3*x+36*A*c*d*e^2
*x+36*B*b*d*e^2*x-24*B*c*d^2*e*x+42*A*b*d*e^2-24*A*c*d^2*e-24*B*b*d^2*e+16*B*c*d^3)/e^4

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 112, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c - 45 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{315 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 111, normalized size = 0.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{5\,e^4}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{7\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 4.18, size = 146, normalized size = 1.16 \begin {gather*} \frac {2 \left (\frac {B c \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A c e + B b e - 3 B c d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- A b d e^{2} + A c d^{2} e + B b d^{2} e - B c 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)*(e*x+d)**(1/2)*(c*x**2+b*x),x)

[Out]

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

________________________________________________________________________________________