### 3.1629 $$\int (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2 \, dx$$

Optimal. Leaf size=129 $-\frac{8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac{12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac{8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac{2 b^4 (d+e x)^{17/2}}{17 e^5}$

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^5) + (12*b^2*(b*d - a*e
)^2*(d + e*x)^(13/2))/(13*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(15/2))/(15*e^5) + (2*b^4*(d + e*x)^(17/2))/(17*
e^5)

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Rubi [A]  time = 0.0707597, 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} $-\frac{8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac{12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac{8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac{2 b^4 (d+e x)^{17/2}}{17 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^5) + (12*b^2*(b*d - a*e
)^2*(d + e*x)^(13/2))/(13*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(15/2))/(15*e^5) + (2*b^4*(d + e*x)^(17/2))/(17*
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)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (d+e x)^{7/2}}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^{9/2}}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^{11/2}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{13/2}}{e^4}+\frac{b^4 (d+e x)^{15/2}}{e^4}\right ) \, dx\\ &=\frac{2 (b d-a e)^4 (d+e x)^{9/2}}{9 e^5}-\frac{8 b (b d-a e)^3 (d+e x)^{11/2}}{11 e^5}+\frac{12 b^2 (b d-a e)^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{15/2}}{15 e^5}+\frac{2 b^4 (d+e x)^{17/2}}{17 e^5}\\ \end{align*}

Mathematica [A]  time = 0.117525, size = 101, normalized size = 0.78 $\frac{2 (d+e x)^{9/2} \left (50490 b^2 (d+e x)^2 (b d-a e)^2-29172 b^3 (d+e x)^3 (b d-a e)-39780 b (d+e x) (b d-a e)^3+12155 (b d-a e)^4+6435 b^4 (d+e x)^4\right )}{109395 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(9/2)*(12155*(b*d - a*e)^4 - 39780*b*(b*d - a*e)^3*(d + e*x) + 50490*b^2*(b*d - a*e)^2*(d + e*x)^
2 - 29172*b^3*(b*d - a*e)*(d + e*x)^3 + 6435*b^4*(d + e*x)^4))/(109395*e^5)

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Maple [A]  time = 0.048, size = 186, normalized size = 1.4 \begin{align*}{\frac{12870\,{x}^{4}{b}^{4}{e}^{4}+58344\,{x}^{3}a{b}^{3}{e}^{4}-6864\,{x}^{3}{b}^{4}d{e}^{3}+100980\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-26928\,{x}^{2}a{b}^{3}d{e}^{3}+3168\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+79560\,x{a}^{3}b{e}^{4}-36720\,x{a}^{2}{b}^{2}d{e}^{3}+9792\,xa{b}^{3}{d}^{2}{e}^{2}-1152\,x{b}^{4}{d}^{3}e+24310\,{a}^{4}{e}^{4}-17680\,{a}^{3}bd{e}^{3}+8160\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-2176\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{109395\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*b^4*e^4*x^4+29172*a*b^3*e^4*x^3-3432*b^4*d*e^3*x^3+50490*a^2*b^2*e^4*x^2-13464*a*
b^3*d*e^3*x^2+1584*b^4*d^2*e^2*x^2+39780*a^3*b*e^4*x-18360*a^2*b^2*d*e^3*x+4896*a*b^3*d^2*e^2*x-576*b^4*d^3*e*
x+12155*a^4*e^4-8840*a^3*b*d*e^3+4080*a^2*b^2*d^2*e^2-1088*a*b^3*d^3*e+128*b^4*d^4)/e^5

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Maxima [A]  time = 1.08627, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} b^{4} - 29172 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 50490 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 39780 \,{\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{11}{2}} + 12155 \,{\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{9}{2}}\right )}}{109395 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

2/109395*(6435*(e*x + d)^(17/2)*b^4 - 29172*(b^4*d - a*b^3*e)*(e*x + d)^(15/2) + 50490*(b^4*d^2 - 2*a*b^3*d*e
+ a^2*b^2*e^2)*(e*x + d)^(13/2) - 39780*(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)^(11/
2) + 12155*(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)^(9/2))/e^5

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Fricas [B]  time = 1.52538, size = 1022, normalized size = 7.92 \begin{align*} \frac{2 \,{\left (6435 \, b^{4} e^{8} x^{8} + 128 \, b^{4} d^{8} - 1088 \, a b^{3} d^{7} e + 4080 \, a^{2} b^{2} d^{6} e^{2} - 8840 \, a^{3} b d^{5} e^{3} + 12155 \, a^{4} d^{4} e^{4} + 1716 \,{\left (13 \, b^{4} d e^{7} + 17 \, a b^{3} e^{8}\right )} x^{7} + 66 \,{\left (401 \, b^{4} d^{2} e^{6} + 1564 \, a b^{3} d e^{7} + 765 \, a^{2} b^{2} e^{8}\right )} x^{6} + 36 \,{\left (303 \, b^{4} d^{3} e^{5} + 3502 \, a b^{3} d^{2} e^{6} + 5100 \, a^{2} b^{2} d e^{7} + 1105 \, a^{3} b e^{8}\right )} x^{5} + 5 \,{\left (7 \, b^{4} d^{4} e^{4} + 10880 \, a b^{3} d^{3} e^{5} + 46716 \, a^{2} b^{2} d^{2} e^{6} + 30056 \, a^{3} b d e^{7} + 2431 \, a^{4} e^{8}\right )} x^{4} - 20 \,{\left (2 \, b^{4} d^{5} e^{3} - 17 \, a b^{3} d^{4} e^{4} - 5406 \, a^{2} b^{2} d^{3} e^{5} - 10166 \, a^{3} b d^{2} e^{6} - 2431 \, a^{4} d e^{7}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{6} e^{2} - 68 \, a b^{3} d^{5} e^{3} + 255 \, a^{2} b^{2} d^{4} e^{4} + 17680 \, a^{3} b d^{3} e^{5} + 12155 \, a^{4} d^{2} e^{6}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{7} e - 136 \, a b^{3} d^{6} e^{2} + 510 \, a^{2} b^{2} d^{5} e^{3} - 1105 \, a^{3} b d^{4} e^{4} - 12155 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/109395*(6435*b^4*e^8*x^8 + 128*b^4*d^8 - 1088*a*b^3*d^7*e + 4080*a^2*b^2*d^6*e^2 - 8840*a^3*b*d^5*e^3 + 1215
5*a^4*d^4*e^4 + 1716*(13*b^4*d*e^7 + 17*a*b^3*e^8)*x^7 + 66*(401*b^4*d^2*e^6 + 1564*a*b^3*d*e^7 + 765*a^2*b^2*
e^8)*x^6 + 36*(303*b^4*d^3*e^5 + 3502*a*b^3*d^2*e^6 + 5100*a^2*b^2*d*e^7 + 1105*a^3*b*e^8)*x^5 + 5*(7*b^4*d^4*
e^4 + 10880*a*b^3*d^3*e^5 + 46716*a^2*b^2*d^2*e^6 + 30056*a^3*b*d*e^7 + 2431*a^4*e^8)*x^4 - 20*(2*b^4*d^5*e^3
- 17*a*b^3*d^4*e^4 - 5406*a^2*b^2*d^3*e^5 - 10166*a^3*b*d^2*e^6 - 2431*a^4*d*e^7)*x^3 + 6*(8*b^4*d^6*e^2 - 68*
a*b^3*d^5*e^3 + 255*a^2*b^2*d^4*e^4 + 17680*a^3*b*d^3*e^5 + 12155*a^4*d^2*e^6)*x^2 - 4*(16*b^4*d^7*e - 136*a*b
^3*d^6*e^2 + 510*a^2*b^2*d^5*e^3 - 1105*a^3*b*d^4*e^4 - 12155*a^4*d^3*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 19.7384, size = 903, normalized size = 7. \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)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Piecewise((2*a**4*d**4*sqrt(d + e*x)/(9*e) + 8*a**4*d**3*x*sqrt(d + e*x)/9 + 4*a**4*d**2*e*x**2*sqrt(d + e*x)/
3 + 8*a**4*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**4*e**3*x**4*sqrt(d + e*x)/9 - 16*a**3*b*d**5*sqrt(d + e*x)/(99*e
**2) + 8*a**3*b*d**4*x*sqrt(d + e*x)/(99*e) + 64*a**3*b*d**3*x**2*sqrt(d + e*x)/33 + 368*a**3*b*d**2*e*x**3*sq
rt(d + e*x)/99 + 272*a**3*b*d*e**2*x**4*sqrt(d + e*x)/99 + 8*a**3*b*e**3*x**5*sqrt(d + e*x)/11 + 32*a**2*b**2*
d**6*sqrt(d + e*x)/(429*e**3) - 16*a**2*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 4*a**2*b**2*d**4*x**2*sqrt(d +
e*x)/(143*e) + 848*a**2*b**2*d**3*x**3*sqrt(d + e*x)/429 + 1832*a**2*b**2*d**2*e*x**4*sqrt(d + e*x)/429 + 480*
a**2*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 12*a**2*b**2*e**3*x**6*sqrt(d + e*x)/13 - 128*a*b**3*d**7*sqrt(d + e
*x)/(6435*e**4) + 64*a*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 16*a*b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) +
8*a*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 1280*a*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 1648*a*b**3*d**2*e*x**
5*sqrt(d + e*x)/715 + 368*a*b**3*d*e**2*x**6*sqrt(d + e*x)/195 + 8*a*b**3*e**3*x**7*sqrt(d + e*x)/15 + 256*b**
4*d**8*sqrt(d + e*x)/(109395*e**5) - 128*b**4*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*b**4*d**6*x**2*sqrt(d +
e*x)/(36465*e**3) - 16*b**4*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*b**4*d**4*x**4*sqrt(d + e*x)/(21879*e) +
2424*b**4*d**3*x**5*sqrt(d + e*x)/12155 + 1604*b**4*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*b**4*d*e**2*x**7*sqr
t(d + e*x)/255 + 2*b**4*e**3*x**8*sqrt(d + e*x)/17, Ne(e, 0)), (d**(7/2)*(a**4*x + 2*a**3*b*x**2 + 2*a**2*b**2
*x**3 + a*b**3*x**4 + b**4*x**5/5), True))

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Giac [B]  time = 1.35548, size = 1729, normalized size = 13.4 \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)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/765765*(204204*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3*b*d^3*e^(-1) + 43758*(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^3*e^(-2) + 9724*(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^3*e^(-3) + 221*(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^3*e^(-
4) + 255255*(x*e + d)^(3/2)*a^4*d^3 + 87516*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2)*a^3*b*d^2*e^(-1) + 43758*(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*d^2*e^(-2) + 2652*(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*d^2*e^(-3) + 255*(693*(x*e + d)^(13/2) - 409
5*(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*d^2*e^(-4) + 153153*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4*d^2 + 29172*(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^3*b*d*e^(-1) + 39
78*(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^2*b^2*d*e^(-2) + 1020*(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)*a*b^3*d*e^(-3)
+ 51*(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)*b^4*d*e^(-4) + 21879*(
15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^4*d + 884*(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^3*b*e^(-1
) + 510*(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)*a^2*b^2*e^(-2) + 68*(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)*a*b^3*e^(-3) + 7*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(1
5/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e
+ d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*b^4*e^(-4) + 2431*(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^4)*e^(-1)