3.1630 \(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)
Optimal. Leaf size=129 \[ -\frac {8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac {12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac {8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5} \]
[Out]
2/7*(-a*e+b*d)^4*(e*x+d)^(7/2)/e^5-8/9*b*(-a*e+b*d)^3*(e*x+d)^(9/2)/e^5+12/11*b^2*(-a*e+b*d)^2*(e*x+d)^(11/2)/
e^5-8/13*b^3*(-a*e+b*d)*(e*x+d)^(13/2)/e^5+2/15*b^4*(e*x+d)^(15/2)/e^5
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Rubi [A] time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00,
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} \[ -\frac {8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac {12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac {8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(2*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^5) + (12*b^2*(b*d - a*e)^
2*(d + e*x)^(11/2))/(11*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^5) + (2*b^4*(d + e*x)^(15/2))/(15*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)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (d+e x)^{5/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{7/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{9/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{11/2}}{e^4}+\frac {b^4 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{9/2}}{9 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{11/2}}{11 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 101, normalized size = 0.78 \[ \frac {2 (d+e x)^{7/2} \left (-13860 b^3 (d+e x)^3 (b d-a e)+24570 b^2 (d+e x)^2 (b d-a e)^2-20020 b (d+e x) (b d-a e)^3+6435 (b d-a e)^4+3003 b^4 (d+e x)^4\right )}{45045 e^5} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(2*(d + e*x)^(7/2)*(6435*(b*d - a*e)^4 - 20020*b*(b*d - a*e)^3*(d + e*x) + 24570*b^2*(b*d - a*e)^2*(d + e*x)^2
- 13860*b^3*(b*d - a*e)*(d + e*x)^3 + 3003*b^4*(d + e*x)^4))/(45045*e^5)
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fricas [B] time = 0.71, size = 377, normalized size = 2.92 \[ \frac {2 \, {\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \, {\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \, {\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} - {\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
[Out]
2/45045*(3003*b^4*e^7*x^7 + 128*b^4*d^7 - 960*a*b^3*d^6*e + 3120*a^2*b^2*d^5*e^2 - 5720*a^3*b*d^4*e^3 + 6435*a
^4*d^3*e^4 + 231*(31*b^4*d*e^6 + 60*a*b^3*e^7)*x^6 + 63*(71*b^4*d^2*e^5 + 540*a*b^3*d*e^6 + 390*a^2*b^2*e^7)*x
^5 + 35*(b^4*d^3*e^4 + 636*a*b^3*d^2*e^5 + 1794*a^2*b^2*d*e^6 + 572*a^3*b*e^7)*x^4 - 5*(8*b^4*d^4*e^3 - 60*a*b
^3*d^3*e^4 - 8814*a^2*b^2*d^2*e^5 - 10868*a^3*b*d*e^6 - 1287*a^4*e^7)*x^3 + 3*(16*b^4*d^5*e^2 - 120*a*b^3*d^4*
e^3 + 390*a^2*b^2*d^3*e^4 + 14300*a^3*b*d^2*e^5 + 6435*a^4*d*e^6)*x^2 - (64*b^4*d^6*e - 480*a*b^3*d^5*e^2 + 15
60*a^2*b^2*d^4*e^3 - 2860*a^3*b*d^3*e^4 - 19305*a^4*d^2*e^5)*x)*sqrt(e*x + d)/e^5
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giac [B] time = 0.24, size = 1277, normalized size = 9.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
[Out]
2/45045*(60060*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d^3*e^(-1) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d
)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*d^3*e^(-2) + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*
e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b^3*d^3*e^(-3) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d +
378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^4*d^3*e^(-4) + 36036*(3*(x*e + d)
^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b*d^2*e^(-1) + 23166*(5*(x*e + d)^(7/2) - 21*(x*e +
d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*b^2*d^2*e^(-2) + 1716*(35*(x*e + d)^(9/2) - 18
0*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^3*d^2*e^(
-3) + 195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 +
1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^4*d^2*e^(-4) + 45045*sqrt(x*e + d)*a^4*d^3 + 45045*((x*e +
d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*d^2 + 15444*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*
d^2 - 35*sqrt(x*e + d)*d^3)*a^3*b*d*e^(-1) + 2574*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^
(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*b^2*d*e^(-2) + 780*(63*(x*e + d)^(11/2) - 385
*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(
x*e + d)*d^5)*a*b^3*d*e^(-3) + 45*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 -
8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^4*
d*e^(-4) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*d + 572*(35*(x*e + d)^(9
/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^3*b
*e^(-1) + 390*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^
3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^2*b^2*e^(-2) + 60*(231*(x*e + d)^(13/2) - 1638*(x*e +
d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^
(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*b^3*e^(-3) + 7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(
x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 1501
5*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^4*e^(-4) + 1287*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^4)*e^(-1)
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maple [A] time = 0.05, size = 186, normalized size = 1.44 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 b^{4} e^{4} x^{4}+13860 a \,b^{3} e^{4} x^{3}-1848 b^{4} d \,e^{3} x^{3}+24570 a^{2} b^{2} e^{4} x^{2}-7560 a \,b^{3} d \,e^{3} x^{2}+1008 b^{4} d^{2} e^{2} x^{2}+20020 a^{3} b \,e^{4} x -10920 a^{2} b^{2} d \,e^{3} x +3360 a \,b^{3} d^{2} e^{2} x -448 b^{4} d^{3} e x +6435 a^{4} e^{4}-5720 a^{3} b d \,e^{3}+3120 a^{2} b^{2} d^{2} e^{2}-960 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{45045 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
2/45045*(e*x+d)^(7/2)*(3003*b^4*e^4*x^4+13860*a*b^3*e^4*x^3-1848*b^4*d*e^3*x^3+24570*a^2*b^2*e^4*x^2-7560*a*b^
3*d*e^3*x^2+1008*b^4*d^2*e^2*x^2+20020*a^3*b*e^4*x-10920*a^2*b^2*d*e^3*x+3360*a*b^3*d^2*e^2*x-448*b^4*d^3*e*x+
6435*a^4*e^4-5720*a^3*b*d*e^3+3120*a^2*b^2*d^2*e^2-960*a*b^3*d^3*e+128*b^4*d^4)/e^5
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maxima [A] time = 1.16, size = 181, normalized size = 1.40 \[ \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{4} - 13860 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 24570 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 20020 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
[Out]
2/45045*(3003*(e*x + d)^(15/2)*b^4 - 13860*(b^4*d - a*b^3*e)*(e*x + d)^(13/2) + 24570*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(e*x + d)^(11/2) - 20020*(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)^(9/2)
+ 6435*(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)^(7/2))/e^5
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mupad [B] time = 0.54, size = 112, normalized size = 0.87 \[ \frac {2\,b^4\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
[Out]
(2*b^4*(d + e*x)^(15/2))/(15*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(13/2))/(13*e^5) + (2*(a*e - b*d)^4*(d +
e*x)^(7/2))/(7*e^5) + (12*b^2*(a*e - b*d)^2*(d + e*x)^(11/2))/(11*e^5) + (8*b*(a*e - b*d)^3*(d + e*x)^(9/2))/(
9*e^5)
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sympy [A] time = 35.26, size = 960, normalized size = 7.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
a**4*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**4*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**4*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
8*a**3*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 16*a**3*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 + 8*a**3*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 + 12*a**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 + 24*a**2*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 + 12*a**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 + 8*a*b**3*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 + 16*a
*b**3*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 + 8*a*b**3*(-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*b**4*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)/11)/e**5 + 4*b**4*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*b**4*(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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