3.1596 \(\int (b+2 c x) (d+e x)^{3/2} (a+b x+c x^2) \, dx\)

Optimal. Leaf size=132 \[ \frac{2 (d+e x)^{7/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^4}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{2 c (d+e x)^{9/2} (2 c d-b e)}{3 e^4}+\frac{4 c^2 (d+e x)^{11/2}}{11 e^4} \]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2))/(5*e^4) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d -
a*e))*(d + e*x)^(7/2))/(7*e^4) - (2*c*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^4) + (4*c^2*(d + e*x)^(11/2))/(11*e^
4)

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Rubi [A]  time = 0.065797, antiderivative size = 132, normalized size of antiderivative = 1., 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} \[ \frac{2 (d+e x)^{7/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^4}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{2 c (d+e x)^{9/2} (2 c d-b e)}{3 e^4}+\frac{4 c^2 (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^(3/2)*(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)^(5/2))/(5*e^4) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d -
a*e))*(d + e*x)^(7/2))/(7*e^4) - (2*c*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^4) + (4*c^2*(d + e*x)^(11/2))/(11*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) (d+e x)^{3/2} \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 ) (d+e x)^{3/2}}{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)^{5/2}}{e^3}-\frac{3 c (2 c d-b e) (d+e x)^{7/2}}{e^3}+\frac{2 c^2 (d+e x)^{9/2}}{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)^{5/2}}{5 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)^{7/2}}{7 e^4}-\frac{2 c (2 c d-b e) (d+e x)^{9/2}}{3 e^4}+\frac{4 c^2 (d+e x)^{11/2}}{11 e^4}\\ \end{align*}

Mathematica [A]  time = 0.139937, size = 109, normalized size = 0.83 \[ \frac{2 (d+e x)^{5/2} \left (11 c e \left (6 a e (5 e x-2 d)+b \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+33 b e^2 (7 a e-2 b d+5 b e x)+c^2 \left (80 d^2 e x-32 d^3-140 d e^2 x^2+210 e^3 x^3\right )\right )}{1155 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(5/2)*(33*b*e^2*(-2*b*d + 7*a*e + 5*b*e*x) + c^2*(-32*d^3 + 80*d^2*e*x - 140*d*e^2*x^2 + 210*e^3*
x^3) + 11*c*e*(6*a*e*(-2*d + 5*e*x) + b*(8*d^2 - 20*d*e*x + 35*e^2*x^2))))/(1155*e^4)

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Maple [A]  time = 0.004, size = 123, normalized size = 0.9 \begin{align*}{\frac{420\,{c}^{2}{x}^{3}{e}^{3}+770\,bc{e}^{3}{x}^{2}-280\,{c}^{2}d{e}^{2}{x}^{2}+660\,ac{e}^{3}x+330\,{b}^{2}{e}^{3}x-440\,bcd{e}^{2}x+160\,{c}^{2}{d}^{2}ex+462\,ab{e}^{3}-264\,acd{e}^{2}-132\,{b}^{2}d{e}^{2}+176\,b{d}^{2}ce-64\,{c}^{2}{d}^{3}}{1155\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a),x)

[Out]

2/1155*(e*x+d)^(5/2)*(210*c^2*e^3*x^3+385*b*c*e^3*x^2-140*c^2*d*e^2*x^2+330*a*c*e^3*x+165*b^2*e^3*x-220*b*c*d*
e^2*x+80*c^2*d^2*e*x+231*a*b*e^3-132*a*c*d*e^2-66*b^2*d*e^2+88*b*c*d^2*e-32*c^2*d^3)/e^4

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Maxima [A]  time = 1.00451, size = 163, normalized size = 1.23 \begin{align*} \frac{2 \,{\left (210 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} - 385 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 165 \,{\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{7}{2}} - 231 \,{\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{5}{2}}\right )}}{1155 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

2/1155*(210*(e*x + d)^(11/2)*c^2 - 385*(2*c^2*d - b*c*e)*(e*x + d)^(9/2) + 165*(6*c^2*d^2 - 6*b*c*d*e + (b^2 +
2*a*c)*e^2)*(e*x + d)^(7/2) - 231*(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)^(5/2))/
e^4

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Fricas [A]  time = 1.33931, size = 505, normalized size = 3.83 \begin{align*} \frac{2 \,{\left (210 \, c^{2} e^{5} x^{5} - 32 \, c^{2} d^{5} + 88 \, b c d^{4} e + 231 \, a b d^{2} e^{3} - 66 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + 35 \,{\left (8 \, c^{2} d e^{4} + 11 \, b c e^{5}\right )} x^{4} + 5 \,{\left (2 \, c^{2} d^{2} e^{3} + 110 \, b c d e^{4} + 33 \,{\left (b^{2} + 2 \, a c\right )} e^{5}\right )} x^{3} - 3 \,{\left (4 \, c^{2} d^{3} e^{2} - 11 \, b c d^{2} e^{3} - 77 \, a b e^{5} - 88 \,{\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{2} +{\left (16 \, c^{2} d^{4} e - 44 \, b c d^{3} e^{2} + 462 \, a b d e^{4} + 33 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{1155 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

2/1155*(210*c^2*e^5*x^5 - 32*c^2*d^5 + 88*b*c*d^4*e + 231*a*b*d^2*e^3 - 66*(b^2 + 2*a*c)*d^3*e^2 + 35*(8*c^2*d
*e^4 + 11*b*c*e^5)*x^4 + 5*(2*c^2*d^2*e^3 + 110*b*c*d*e^4 + 33*(b^2 + 2*a*c)*e^5)*x^3 - 3*(4*c^2*d^3*e^2 - 11*
b*c*d^2*e^3 - 77*a*b*e^5 - 88*(b^2 + 2*a*c)*d*e^4)*x^2 + (16*c^2*d^4*e - 44*b*c*d^3*e^2 + 462*a*b*d*e^4 + 33*(
b^2 + 2*a*c)*d^2*e^3)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 15.9213, size = 457, normalized size = 3.46 \begin{align*} a b d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a b \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{4 a c d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 a c \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 b^{2} d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 b^{2} \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{6 b c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{6 b c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{4 c^{2} d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{4 c^{2} \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**(3/2)*(c*x**2+b*x+a),x)

[Out]

a*b*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*b*(-d*(d + e*x)**(3/2)/3 + (d +
e*x)**(5/2)/5)/e + 4*a*c*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*a*c*(d**2*(d + e*x)**(3/2)/3
- 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*b**2*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e
**2 + 2*b**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*b*c*d*(d**2*(d +
e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 6*b*c*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*
(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 4*c**2*d*(-d**3*(d + e*x)**(3/2)/3 +
3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 4*c**2*(d**4*(d + e*x)**(3/2)/
3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**
4

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Giac [B]  time = 1.50382, size = 548, normalized size = 4.15 \begin{align*} \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b^{2} d e^{\left (-1\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a c d e^{\left (-1\right )} + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b c d e^{\left (-2\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} c^{2} d e^{\left (-3\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a b d + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b^{2} e^{\left (-1\right )} + 66 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a c e^{\left (-1\right )} + 33 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b c e^{\left (-2\right )} + 2 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} c^{2} e^{\left (-3\right )} + 231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b^2*d*e^(-1) + 462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2
)*d)*a*c*d*e^(-1) + 99*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b*c*d*e^(-2) + 22*
(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*c^2*d*e^(-3)
+ 1155*(x*e + d)^(3/2)*a*b*d + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*e^(
-1) + 66*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*c*e^(-1) + 33*(35*(x*e + d)^(9
/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b*c*e^(-2) + 2*(315*(x*e + d)
^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*
d^4)*c^2*e^(-3) + 231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b)*e^(-1)