3.15.5 \(\int (b+2 c x) \sqrt {d+e x} (a+b x+c x^2) \, dx\)

Optimal. Leaf size=132 \[ \frac {2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {6 c (d+e x)^{7/2} (2 c d-b e)}{7 e^4}+\frac {4 c^2 (d+e x)^{9/2}}{9 e^4} \]

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Rubi [A]  time = 0.07, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {6 c (d+e x)^{7/2} (2 c d-b e)}{7 e^4}+\frac {4 c^2 (d+e x)^{9/2}}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2))/(3*e^4) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d -
a*e))*(d + e*x)^(5/2))/(5*e^4) - (6*c*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^4) + (4*c^2*(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 (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^3}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{e^3}-\frac {3 c (2 c d-b e) (d+e x)^{5/2}}{e^3}+\frac {2 c^2 (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^4}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{5/2}}{5 e^4}-\frac {6 c (2 c d-b e) (d+e x)^{7/2}}{7 e^4}+\frac {4 c^2 (d+e x)^{9/2}}{9 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 110, normalized size = 0.83 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (3 c e \left (14 a e (3 e x-2 d)+3 b \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+21 b e^2 (5 a e-2 b d+3 b e x)+c^2 \left (-32 d^3+48 d^2 e x-60 d e^2 x^2+70 e^3 x^3\right )\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(21*b*e^2*(-2*b*d + 5*a*e + 3*b*e*x) + c^2*(-32*d^3 + 48*d^2*e*x - 60*d*e^2*x^2 + 70*e^3*x^
3) + 3*c*e*(14*a*e*(-2*d + 3*e*x) + 3*b*(8*d^2 - 12*d*e*x + 15*e^2*x^2))))/(315*e^4)

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IntegrateAlgebraic [A]  time = 0.08, size = 143, normalized size = 1.08 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (105 a b e^3+126 a c e^2 (d+e x)-210 a c d e^2+63 b^2 e^2 (d+e x)-105 b^2 d e^2+315 b c d^2 e-378 b c d e (d+e x)+135 b c e (d+e x)^2-210 c^2 d^3+378 c^2 d^2 (d+e x)-270 c^2 d (d+e x)^2+70 c^2 (d+e x)^3\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-210*c^2*d^3 + 315*b*c*d^2*e - 105*b^2*d*e^2 - 210*a*c*d*e^2 + 105*a*b*e^3 + 378*c^2*d^2*(
d + e*x) - 378*b*c*d*e*(d + e*x) + 63*b^2*e^2*(d + e*x) + 126*a*c*e^2*(d + e*x) - 270*c^2*d*(d + e*x)^2 + 135*
b*c*e*(d + e*x)^2 + 70*c^2*(d + e*x)^3))/(315*e^4)

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fricas [A]  time = 0.40, size = 167, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (70 \, c^{2} e^{4} x^{4} - 32 \, c^{2} d^{4} + 72 \, b c d^{3} e + 105 \, a b d e^{3} - 42 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 5 \, {\left (2 \, c^{2} d e^{3} + 27 \, b c e^{4}\right )} x^{3} - 3 \, {\left (4 \, c^{2} d^{2} e^{2} - 9 \, b c d e^{3} - 21 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + {\left (16 \, c^{2} d^{3} e - 36 \, b c d^{2} e^{2} + 105 \, a b e^{4} + 21 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(70*c^2*e^4*x^4 - 32*c^2*d^4 + 72*b*c*d^3*e + 105*a*b*d*e^3 - 42*(b^2 + 2*a*c)*d^2*e^2 + 5*(2*c^2*d*e^3
+ 27*b*c*e^4)*x^3 - 3*(4*c^2*d^2*e^2 - 9*b*c*d*e^3 - 21*(b^2 + 2*a*c)*e^4)*x^2 + (16*c^2*d^3*e - 36*b*c*d^2*e^
2 + 105*a*b*e^4 + 21*(b^2 + 2*a*c)*d*e^3)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 0.27, size = 400, normalized size = 3.03 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b^{2} d e^{\left (-1\right )} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a c d e^{\left (-1\right )} + 63 \, {\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 c d e^{\left (-2\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 )} c^{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^{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 c e^{\left (-1\right )} + 27 \, {\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 e^{\left (-2\right )} + 2 \, {\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 )} c^{2} e^{\left (-3\right )} + 315 \, \sqrt {x e + d} a b d + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a b\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^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^2*d*e^(-1) + 210*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*c*
d*e^(-1) + 63*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b*c*d*e^(-2) + 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)*c^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^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*c*e^(-1) + 27*(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*e^(-2) + 2*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 4
20*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c^2*e^(-3) + 315*sqrt(x*e + d)*a*b*d + 105*((x*e + d)^(3/2) -
3*sqrt(x*e + d)*d)*a*b)*e^(-1)

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maple [A]  time = 0.05, size = 123, normalized size = 0.93 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (70 c^{2} x^{3} e^{3}+135 b c \,e^{3} x^{2}-60 c^{2} d \,e^{2} x^{2}+126 a c \,e^{3} x +63 b^{2} e^{3} x -108 b c d \,e^{2} x +48 c^{2} d^{2} e x +105 a b \,e^{3}-84 a c d \,e^{2}-42 b^{2} d \,e^{2}+72 b c \,d^{2} e -32 c^{2} d^{3}\right )}{315 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(e*x+d)^(3/2)*(70*c^2*e^3*x^3+135*b*c*e^3*x^2-60*c^2*d*e^2*x^2+126*a*c*e^3*x+63*b^2*e^3*x-108*b*c*d*e^2*
x+48*c^2*d^2*e*x+105*a*b*e^3-84*a*c*d*e^2-42*b^2*d*e^2+72*b*c*d^2*e-32*c^2*d^3)/e^4

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maxima [A]  time = 0.52, size = 121, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (70 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{2} - 135 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 63 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\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((2*c*x+b)*(c*x^2+b*x+a)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/315*(70*(e*x + d)^(9/2)*c^2 - 135*(2*c^2*d - b*c*e)*(e*x + d)^(7/2) + 63*(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a
*c)*e^2)*(e*x + d)^(5/2) - 105*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)^(3/2))/e^4

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mupad [B]  time = 0.07, size = 118, normalized size = 0.89 \begin {gather*} \frac {4\,c^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{5\,e^4}-\frac {\left (12\,c^2\,d-6\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c^2*(d + e*x)^(9/2))/(9*e^4) + ((d + e*x)^(5/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(5*e^4)
- ((12*c^2*d - 6*b*c*e)*(d + e*x)^(7/2))/(7*e^4) + (2*(b*e - 2*c*d)*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e))/(
3*e^4)

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sympy [A]  time = 5.14, size = 155, normalized size = 1.17 \begin {gather*} \frac {2 \left (\frac {2 c^{2} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (3 b c e - 6 c^{2} d\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{5 e^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 3 b c d^{2} e - 2 c^{2} d^{3}\right )}{3 e^{3}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*c**2*(d + e*x)**(9/2)/(9*e**3) + (d + e*x)**(7/2)*(3*b*c*e - 6*c**2*d)/(7*e**3) + (d + e*x)**(5/2)*(2*a*c
*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(5*e**3) + (d + e*x)**(3/2)*(a*b*e**3 - 2*a*c*d*e**2 - b**2*d*e**
2 + 3*b*c*d**2*e - 2*c**2*d**3)/(3*e**3))/e

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