3.1595 \(\int (b+2 c x) (d+e x)^{5/2} (a+b x+c x^2) \, dx\)

Optimal. Leaf size=132 \[ \frac{2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^4}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^4}-\frac{6 c (d+e x)^{11/2} (2 c d-b e)}{11 e^4}+\frac{4 c^2 (d+e x)^{13/2}}{13 e^4} \]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(7/2))/(7*e^4) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d -
a*e))*(d + e*x)^(9/2))/(9*e^4) - (6*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^4) + (4*c^2*(d + e*x)^(13/2))/(13*
e^4)

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Rubi [A]  time = 0.0822013, 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} \[ \frac{2 (d+e x)^{9/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{9 e^4}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{7 e^4}-\frac{6 c (d+e x)^{11/2} (2 c d-b e)}{11 e^4}+\frac{4 c^2 (d+e x)^{13/2}}{13 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^(5/2)*(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)^(7/2))/(7*e^4) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d -
a*e))*(d + e*x)^(9/2))/(9*e^4) - (6*c*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^4) + (4*c^2*(d + e*x)^(13/2))/(13*
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)^{5/2} \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)^{5/2}}{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)^{7/2}}{e^3}-\frac{3 c (2 c d-b e) (d+e x)^{9/2}}{e^3}+\frac{2 c^2 (d+e x)^{11/2}}{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)^{7/2}}{7 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)^{9/2}}{9 e^4}-\frac{6 c (2 c d-b e) (d+e x)^{11/2}}{11 e^4}+\frac{4 c^2 (d+e x)^{13/2}}{13 e^4}\\ \end{align*}

Mathematica [A]  time = 0.150667, size = 111, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (13 c e \left (22 a e (7 e x-2 d)+3 b \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+143 b e^2 (9 a e-2 b d+7 b e x)-6 c^2 \left (-56 d^2 e x+16 d^3+126 d e^2 x^2-231 e^3 x^3\right )\right )}{9009 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(7/2)*(143*b*e^2*(-2*b*d + 9*a*e + 7*b*e*x) - 6*c^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^
3*x^3) + 13*c*e*(22*a*e*(-2*d + 7*e*x) + 3*b*(8*d^2 - 28*d*e*x + 63*e^2*x^2))))/(9009*e^4)

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Maple [A]  time = 0.004, size = 123, normalized size = 0.9 \begin{align*}{\frac{2772\,{c}^{2}{x}^{3}{e}^{3}+4914\,bc{e}^{3}{x}^{2}-1512\,{c}^{2}d{e}^{2}{x}^{2}+4004\,ac{e}^{3}x+2002\,{b}^{2}{e}^{3}x-2184\,bcd{e}^{2}x+672\,{c}^{2}{d}^{2}ex+2574\,ab{e}^{3}-1144\,acd{e}^{2}-572\,{b}^{2}d{e}^{2}+624\,b{d}^{2}ce-192\,{c}^{2}{d}^{3}}{9009\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a),x)

[Out]

2/9009*(e*x+d)^(7/2)*(1386*c^2*e^3*x^3+2457*b*c*e^3*x^2-756*c^2*d*e^2*x^2+2002*a*c*e^3*x+1001*b^2*e^3*x-1092*b
*c*d*e^2*x+336*c^2*d^2*e*x+1287*a*b*e^3-572*a*c*d*e^2-286*b^2*d*e^2+312*b*c*d^2*e-96*c^2*d^3)/e^4

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Maxima [A]  time = 0.990196, size = 163, normalized size = 1.23 \begin{align*} \frac{2 \,{\left (1386 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} - 2457 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 1001 \,{\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{9}{2}} - 1287 \,{\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{7}{2}}\right )}}{9009 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

2/9009*(1386*(e*x + d)^(13/2)*c^2 - 2457*(2*c^2*d - b*c*e)*(e*x + d)^(11/2) + 1001*(6*c^2*d^2 - 6*b*c*d*e + (b
^2 + 2*a*c)*e^2)*(e*x + d)^(9/2) - 1287*(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)^(7
/2))/e^4

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Fricas [B]  time = 1.33152, size = 648, normalized size = 4.91 \begin{align*} \frac{2 \,{\left (1386 \, c^{2} e^{6} x^{6} - 96 \, c^{2} d^{6} + 312 \, b c d^{5} e + 1287 \, a b d^{3} e^{3} - 286 \,{\left (b^{2} + 2 \, a c\right )} d^{4} e^{2} + 189 \,{\left (18 \, c^{2} d e^{5} + 13 \, b c e^{6}\right )} x^{5} + 7 \,{\left (318 \, c^{2} d^{2} e^{4} + 897 \, b c d e^{5} + 143 \,{\left (b^{2} + 2 \, a c\right )} e^{6}\right )} x^{4} +{\left (30 \, c^{2} d^{3} e^{3} + 4407 \, b c d^{2} e^{4} + 1287 \, a b e^{6} + 2717 \,{\left (b^{2} + 2 \, a c\right )} d e^{5}\right )} x^{3} - 3 \,{\left (12 \, c^{2} d^{4} e^{2} - 39 \, b c d^{3} e^{3} - 1287 \, a b d e^{5} - 715 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{2} +{\left (48 \, c^{2} d^{5} e - 156 \, b c d^{4} e^{2} + 3861 \, a b d^{2} e^{4} + 143 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

2/9009*(1386*c^2*e^6*x^6 - 96*c^2*d^6 + 312*b*c*d^5*e + 1287*a*b*d^3*e^3 - 286*(b^2 + 2*a*c)*d^4*e^2 + 189*(18
*c^2*d*e^5 + 13*b*c*e^6)*x^5 + 7*(318*c^2*d^2*e^4 + 897*b*c*d*e^5 + 143*(b^2 + 2*a*c)*e^6)*x^4 + (30*c^2*d^3*e
^3 + 4407*b*c*d^2*e^4 + 1287*a*b*e^6 + 2717*(b^2 + 2*a*c)*d*e^5)*x^3 - 3*(12*c^2*d^4*e^2 - 39*b*c*d^3*e^3 - 12
87*a*b*d*e^5 - 715*(b^2 + 2*a*c)*d^2*e^4)*x^2 + (48*c^2*d^5*e - 156*b*c*d^4*e^2 + 3861*a*b*d^2*e^4 + 143*(b^2
+ 2*a*c)*d^3*e^3)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 4.93978, size = 643, normalized size = 4.87 \begin{align*} \begin{cases} \frac{2 a b d^{3} \sqrt{d + e x}}{7 e} + \frac{6 a b d^{2} x \sqrt{d + e x}}{7} + \frac{6 a b d e x^{2} \sqrt{d + e x}}{7} + \frac{2 a b e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{8 a c d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{4 a c d^{3} x \sqrt{d + e x}}{63 e} + \frac{20 a c d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{76 a c d e x^{3} \sqrt{d + e x}}{63} + \frac{4 a c e^{2} x^{4} \sqrt{d + e x}}{9} - \frac{4 b^{2} d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 b^{2} d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 b^{2} d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 b^{2} d e x^{3} \sqrt{d + e x}}{63} + \frac{2 b^{2} e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 b c d^{5} \sqrt{d + e x}}{231 e^{3}} - \frac{8 b c d^{4} x \sqrt{d + e x}}{231 e^{2}} + \frac{2 b c d^{3} x^{2} \sqrt{d + e x}}{77 e} + \frac{226 b c d^{2} x^{3} \sqrt{d + e x}}{231} + \frac{46 b c d e x^{4} \sqrt{d + e x}}{33} + \frac{6 b c e^{2} x^{5} \sqrt{d + e x}}{11} - \frac{64 c^{2} d^{6} \sqrt{d + e x}}{3003 e^{4}} + \frac{32 c^{2} d^{5} x \sqrt{d + e x}}{3003 e^{3}} - \frac{8 c^{2} d^{4} x^{2} \sqrt{d + e x}}{1001 e^{2}} + \frac{20 c^{2} d^{3} x^{3} \sqrt{d + e x}}{3003 e} + \frac{212 c^{2} d^{2} x^{4} \sqrt{d + e x}}{429} + \frac{108 c^{2} d e x^{5} \sqrt{d + e x}}{143} + \frac{4 c^{2} e^{2} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (a b x + a c x^{2} + \frac{b^{2} x^{2}}{2} + b c x^{3} + \frac{c^{2} x^{4}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**(5/2)*(c*x**2+b*x+a),x)

[Out]

Piecewise((2*a*b*d**3*sqrt(d + e*x)/(7*e) + 6*a*b*d**2*x*sqrt(d + e*x)/7 + 6*a*b*d*e*x**2*sqrt(d + e*x)/7 + 2*
a*b*e**2*x**3*sqrt(d + e*x)/7 - 8*a*c*d**4*sqrt(d + e*x)/(63*e**2) + 4*a*c*d**3*x*sqrt(d + e*x)/(63*e) + 20*a*
c*d**2*x**2*sqrt(d + e*x)/21 + 76*a*c*d*e*x**3*sqrt(d + e*x)/63 + 4*a*c*e**2*x**4*sqrt(d + e*x)/9 - 4*b**2*d**
4*sqrt(d + e*x)/(63*e**2) + 2*b**2*d**3*x*sqrt(d + e*x)/(63*e) + 10*b**2*d**2*x**2*sqrt(d + e*x)/21 + 38*b**2*
d*e*x**3*sqrt(d + e*x)/63 + 2*b**2*e**2*x**4*sqrt(d + e*x)/9 + 16*b*c*d**5*sqrt(d + e*x)/(231*e**3) - 8*b*c*d*
*4*x*sqrt(d + e*x)/(231*e**2) + 2*b*c*d**3*x**2*sqrt(d + e*x)/(77*e) + 226*b*c*d**2*x**3*sqrt(d + e*x)/231 + 4
6*b*c*d*e*x**4*sqrt(d + e*x)/33 + 6*b*c*e**2*x**5*sqrt(d + e*x)/11 - 64*c**2*d**6*sqrt(d + e*x)/(3003*e**4) +
32*c**2*d**5*x*sqrt(d + e*x)/(3003*e**3) - 8*c**2*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 20*c**2*d**3*x**3*sqrt
(d + e*x)/(3003*e) + 212*c**2*d**2*x**4*sqrt(d + e*x)/429 + 108*c**2*d*e*x**5*sqrt(d + e*x)/143 + 4*c**2*e**2*
x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(5/2)*(a*b*x + a*c*x**2 + b**2*x**2/2 + b*c*x**3 + c**2*x**4/2), True))

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Giac [B]  time = 1.59413, size = 968, normalized size = 7.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b^2*d^2*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)
^(3/2)*d)*a*c*d^2*e^(-1) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b*c*d^2*e
^(-2) + 286*(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^2*e^(-3) + 15015*(x*e + d)^(3/2)*a*b*d^2 + 858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^
(3/2)*d^2)*b^2*d*e^(-1) + 1716*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*c*d*e^(-
1) + 858*(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*
d*e^(-2) + 52*(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*d*e^(-3) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*d + 143*(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^2*e^(-1) + 286
*(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)*a*c*e^(-1) +
39*(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 + 115
5*(x*e + d)^(3/2)*d^4)*b*c*e^(-2) + 10*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)
*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*c^2*e^(-3) + 429*(15*(
x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b)*e^(-1)