3.2275 $$\int (d+e x)^{5/2} (a+b x+c x^2)^2 \, dx$$

Optimal. Leaf size=166 $\frac{2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac{4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}$

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(9/
2))/(9*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(11/2))/(11*e^5) - (4*c*(2*c*d - b*e)*(
d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*e^5)

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Rubi [A]  time = 0.092938, antiderivative size = 166, normalized size of antiderivative = 1., 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 = {698} $\frac{2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac{4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(9/
2))/(9*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(11/2))/(11*e^5) - (4*c*(2*c*d - b*e)*(
d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*e^5)

Rule 698

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

Rubi steps

\begin{align*} \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{e^4}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{e^4}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{9/2}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{11/2}}{e^4}+\frac{c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac{2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{11/2}}{11 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}\\ \end{align*}

Mathematica [A]  time = 0.163742, size = 173, normalized size = 1.04 $\frac{2 (d+e x)^{7/2} \left (65 e^2 \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-10 c e \left (3 b \left (-56 d^2 e x+16 d^3+126 d e^2 x^2-231 e^3 x^3\right )-13 a e \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+c^2 \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 65*e^2*(9
9*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 10*c*e*(-13*a*e*(8*d^2 - 28*d*e*x
+ 63*e^2*x^2) + 3*b*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3))))/(45045*e^5)

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Maple [A]  time = 0.048, size = 194, normalized size = 1.2 \begin{align*}{\frac{6006\,{c}^{2}{x}^{4}{e}^{4}+13860\,bc{e}^{4}{x}^{3}-3696\,{c}^{2}d{e}^{3}{x}^{3}+16380\,ac{e}^{4}{x}^{2}+8190\,{b}^{2}{e}^{4}{x}^{2}-7560\,bcd{e}^{3}{x}^{2}+2016\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+20020\,ab{e}^{4}x-7280\,acd{e}^{3}x-3640\,{b}^{2}d{e}^{3}x+3360\,bc{d}^{2}{e}^{2}x-896\,{c}^{2}{d}^{3}ex+12870\,{a}^{2}{e}^{4}-5720\,abd{e}^{3}+2080\,ac{d}^{2}{e}^{2}+1040\,{b}^{2}{d}^{2}{e}^{2}-960\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(7/2)*(3003*c^2*e^4*x^4+6930*b*c*e^4*x^3-1848*c^2*d*e^3*x^3+8190*a*c*e^4*x^2+4095*b^2*e^4*x^2-
3780*b*c*d*e^3*x^2+1008*c^2*d^2*e^2*x^2+10010*a*b*e^4*x-3640*a*c*d*e^3*x-1820*b^2*d*e^3*x+1680*b*c*d^2*e^2*x-4
48*c^2*d^3*e*x+6435*a^2*e^4-2860*a*b*d*e^3+1040*a*c*d^2*e^2+520*b^2*d^2*e^2-480*b*c*d^3*e+128*c^2*d^4)/e^5

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Maxima [A]  time = 0.983419, size = 238, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{2} - 6930 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 4095 \,{\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{11}{2}} - 10010 \,{\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{9}{2}} + 6435 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^2 - 6930*(2*c^2*d - b*c*e)*(e*x + d)^(13/2) + 4095*(6*c^2*d^2 - 6*b*c*d*e + (
b^2 + 2*a*c)*e^2)*(e*x + d)^(11/2) - 10010*(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)
^(9/2) + 6435*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*(e*x + d)^(7/2))/e^5

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Fricas [B]  time = 2.35495, size = 860, normalized size = 5.18 \begin{align*} \frac{2 \,{\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e - 2860 \, a b d^{4} e^{3} + 6435 \, a^{2} d^{3} e^{4} + 520 \,{\left (b^{2} + 2 \, a c\right )} d^{5} e^{2} + 231 \,{\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \,{\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \,{\left (b^{2} + 2 \, a c\right )} e^{7}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 286 \, a b e^{7} + 299 \,{\left (b^{2} + 2 \, a c\right )} d e^{6}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 5434 \, a b d e^{6} - 1287 \, a^{2} e^{7} - 1469 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 7150 \, a b d^{2} e^{5} + 6435 \, a^{2} d e^{6} + 65 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{2} -{\left (64 \, c^{2} d^{6} e - 240 \, b c d^{5} e^{2} - 1430 \, a b d^{3} e^{4} - 19305 \, a^{2} d^{2} e^{5} + 260 \,{\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*c^2*e^7*x^7 + 128*c^2*d^7 - 480*b*c*d^6*e - 2860*a*b*d^4*e^3 + 6435*a^2*d^3*e^4 + 520*(b^2 + 2*a
*c)*d^5*e^2 + 231*(31*c^2*d*e^6 + 30*b*c*e^7)*x^6 + 63*(71*c^2*d^2*e^5 + 270*b*c*d*e^6 + 65*(b^2 + 2*a*c)*e^7)
*x^5 + 35*(c^2*d^3*e^4 + 318*b*c*d^2*e^5 + 286*a*b*e^7 + 299*(b^2 + 2*a*c)*d*e^6)*x^4 - 5*(8*c^2*d^4*e^3 - 30*
b*c*d^3*e^4 - 5434*a*b*d*e^6 - 1287*a^2*e^7 - 1469*(b^2 + 2*a*c)*d^2*e^5)*x^3 + 3*(16*c^2*d^5*e^2 - 60*b*c*d^4
*e^3 + 7150*a*b*d^2*e^5 + 6435*a^2*d*e^6 + 65*(b^2 + 2*a*c)*d^3*e^4)*x^2 - (64*c^2*d^6*e - 240*b*c*d^5*e^2 - 1
430*a*b*d^3*e^4 - 19305*a^2*d^2*e^5 + 260*(b^2 + 2*a*c)*d^4*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 34.4345, size = 1129, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**2*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
4*a*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*a*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x
)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*a*b*(-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**2 + 4*a*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d
+ e*x)**(7/2)/7)/e**3 + 8*a*c*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**3 + 4*a*c*(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**3 + 2*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d +
e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 4*b**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**3 + 2*b**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**3 + 4*b*c*d**2*(-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 + 8*b*c*d*(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 + 4*b*c*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d
**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**4 + 2*c**2*d**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)/1
1)/e**5 + 4*c**2*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d
+ e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 2*c**2*(d**6*(d + e*x)**(3/2)/3 - 6
*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)
/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

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Giac [B]  time = 1.15802, size = 1353, normalized size = 8.15 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*d^2*e^(-1) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*d^2*e^(-2) + 858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e
+ d)^(3/2)*d^2)*a*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 - 1
05*(x*e + d)^(3/2)*d^3)*b*c*d^2*e^(-3) + 13*(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^2*e^(-4) + 15015*(x*e + d)^(3/2)*a^2*d^2
+ 1716*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b*d*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)*b^2*d*e^(-2) + 572*(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*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)*b*c*d*e^(-3) + 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*d*e^(-4) + 6006*(3*(x*e
+ d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*d + 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*b*e^(-1) + 13*(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)*b^2*e^(-2) + 26*(315*(x*e + d)^(11/2) - 15
40*(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)*a*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)*b*c*e^(-3) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d
)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e
+ d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c^2*e^(-4) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(
x*e + d)^(3/2)*d^2)*a^2)*e^(-1)