### 3.1631 $$\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2 \, dx$$

Optimal. Leaf size=129 $-\frac{8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac{8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5}$

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2))/(5*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^5) + (4*b^2*(b*d - a*e)^2
*(d + e*x)^(9/2))/(3*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^5) + (2*b^4*(d + e*x)^(13/2))/(13*e^5)

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Rubi [A]  time = 0.0428612, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {27, 43} $-\frac{8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac{8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac{2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2))/(5*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^5) + (4*b^2*(b*d - a*e)^2
*(d + e*x)^(9/2))/(3*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^5) + (2*b^4*(d + e*x)^(13/2))/(13*e^5)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (d+e x)^{3/2}}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^{5/2}}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{7/2}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{9/2}}{e^4}+\frac{b^4 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^4 (d+e x)^{5/2}}{5 e^5}-\frac{8 b (b d-a e)^3 (d+e x)^{7/2}}{7 e^5}+\frac{4 b^2 (b d-a e)^2 (d+e x)^{9/2}}{3 e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{11/2}}{11 e^5}+\frac{2 b^4 (d+e x)^{13/2}}{13 e^5}\\ \end{align*}

Mathematica [A]  time = 0.0845709, size = 101, normalized size = 0.78 $\frac{2 (d+e x)^{5/2} \left (10010 b^2 (d+e x)^2 (b d-a e)^2-5460 b^3 (d+e x)^3 (b d-a e)-8580 b (d+e x) (b d-a e)^3+3003 (b d-a e)^4+1155 b^4 (d+e x)^4\right )}{15015 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(3003*(b*d - a*e)^4 - 8580*b*(b*d - a*e)^3*(d + e*x) + 10010*b^2*(b*d - a*e)^2*(d + e*x)^2
- 5460*b^3*(b*d - a*e)*(d + e*x)^3 + 1155*b^4*(d + e*x)^4))/(15015*e^5)

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Maple [A]  time = 0.046, size = 186, normalized size = 1.4 \begin{align*}{\frac{2310\,{x}^{4}{b}^{4}{e}^{4}+10920\,{x}^{3}a{b}^{3}{e}^{4}-1680\,{x}^{3}{b}^{4}d{e}^{3}+20020\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-7280\,{x}^{2}a{b}^{3}d{e}^{3}+1120\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+17160\,x{a}^{3}b{e}^{4}-11440\,x{a}^{2}{b}^{2}d{e}^{3}+4160\,xa{b}^{3}{d}^{2}{e}^{2}-640\,x{b}^{4}{d}^{3}e+6006\,{a}^{4}{e}^{4}-6864\,{a}^{3}bd{e}^{3}+4576\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-1664\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{15015\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(e*x+d)^(5/2)*(1155*b^4*e^4*x^4+5460*a*b^3*e^4*x^3-840*b^4*d*e^3*x^3+10010*a^2*b^2*e^4*x^2-3640*a*b^3*
d*e^3*x^2+560*b^4*d^2*e^2*x^2+8580*a^3*b*e^4*x-5720*a^2*b^2*d*e^3*x+2080*a*b^3*d^2*e^2*x-320*b^4*d^3*e*x+3003*
a^4*e^4-3432*a^3*b*d*e^3+2288*a^2*b^2*d^2*e^2-832*a*b^3*d^3*e+128*b^4*d^4)/e^5

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Maxima [A]  time = 1.10382, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (1155 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{4} - 5460 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 8580 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 3003 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{15015 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*(e*x + d)^(13/2)*b^4 - 5460*(b^4*d - a*b^3*e)*(e*x + d)^(11/2) + 10010*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(e*x + d)^(9/2) - 8580*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d)^(7/2) +
3003*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(e*x + d)^(5/2))/e^5

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Fricas [B]  time = 1.49044, size = 702, normalized size = 5.44 \begin{align*} \frac{2 \,{\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \,{\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \,{\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \,{\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \,{\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*b^4*e^6*x^6 + 128*b^4*d^6 - 832*a*b^3*d^5*e + 2288*a^2*b^2*d^4*e^2 - 3432*a^3*b*d^3*e^3 + 3003*a
^4*d^2*e^4 + 210*(7*b^4*d*e^5 + 26*a*b^3*e^6)*x^5 + 35*(b^4*d^2*e^4 + 208*a*b^3*d*e^5 + 286*a^2*b^2*e^6)*x^4 -
20*(2*b^4*d^3*e^3 - 13*a*b^3*d^2*e^4 - 715*a^2*b^2*d*e^5 - 429*a^3*b*e^6)*x^3 + 3*(16*b^4*d^4*e^2 - 104*a*b^3
*d^3*e^3 + 286*a^2*b^2*d^2*e^4 + 4576*a^3*b*d*e^5 + 1001*a^4*e^6)*x^2 - 2*(32*b^4*d^5*e - 208*a*b^3*d^4*e^2 +
572*a^2*b^2*d^3*e^3 - 858*a^3*b*d^2*e^4 - 3003*a^4*d*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 18.9131, size = 559, normalized size = 4.33 \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)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**4*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**4*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 8*a**3*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*a**3*b*(d**2*(d + e*x)*
*(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12*a**2*b**2*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 + 12*a**2*b**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**3 + 8*a*b**3*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 + 8*a*b**3*(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 + 2*b**4*
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**5 + 2*b**4*(-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

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Giac [B]  time = 1.25953, size = 675, normalized size = 5.23 \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)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/45045*(12012*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*b*d*e^(-1) + 2574*(15*(x*e + d)^(7/2) - 42*(x*e +
d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*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*b^3*d*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)*b^4*d*e^(-4) + 15015*(x
*e + d)^(3/2)*a^4*d + 1716*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*b*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)*a^2*b^2*
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)*a*b^3*e^(-3) + 5*(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^4*e^(-4) +
3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4)*e^(-1)