3.17.38 \(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1638]

Optimal. Leaf size=185 \[ \frac {2 (b d-a e)^6 (d+e x)^{7/2}}{7 e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{9/2}}{3 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{11/2}}{11 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{13/2}}{13 e^7}+\frac {2 b^4 (b d-a e)^2 (d+e x)^{15/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{17/2}}{17 e^7}+\frac {2 b^6 (d+e x)^{19/2}}{19 e^7} \]

[Out]

2/7*(-a*e+b*d)^6*(e*x+d)^(7/2)/e^7-4/3*b*(-a*e+b*d)^5*(e*x+d)^(9/2)/e^7+30/11*b^2*(-a*e+b*d)^4*(e*x+d)^(11/2)/
e^7-40/13*b^3*(-a*e+b*d)^3*(e*x+d)^(13/2)/e^7+2*b^4*(-a*e+b*d)^2*(e*x+d)^(15/2)/e^7-12/17*b^5*(-a*e+b*d)*(e*x+
d)^(17/2)/e^7+2/19*b^6*(e*x+d)^(19/2)/e^7

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Rubi [A]
time = 0.04, antiderivative size = 185, 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, 45} \begin {gather*} -\frac {12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac {2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac {40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac {30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac {4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac {2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac {2 b^6 (d+e x)^{19/2}}{19 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(7/2))/(7*e^7) - (4*b*(b*d - a*e)^5*(d + e*x)^(9/2))/(3*e^7) + (30*b^2*(b*d - a*e)^
4*(d + e*x)^(11/2))/(11*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(13/2))/(13*e^7) + (2*b^4*(b*d - a*e)^2*(d + e*
x)^(15/2))/e^7 - (12*b^5*(b*d - a*e)*(d + e*x)^(17/2))/(17*e^7) + (2*b^6*(d + e*x)^(19/2))/(19*e^7)

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 45

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 )^3 \, dx &=\int (a+b x)^6 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (d+e x)^{5/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{7/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{9/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{13/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{15/2}}{e^6}+\frac {b^6 (d+e x)^{17/2}}{e^6}\right ) \, dx\\ &=\frac {2 (b d-a e)^6 (d+e x)^{7/2}}{7 e^7}-\frac {4 b (b d-a e)^5 (d+e x)^{9/2}}{3 e^7}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{11/2}}{11 e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{13/2}}{13 e^7}+\frac {2 b^4 (b d-a e)^2 (d+e x)^{15/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{17/2}}{17 e^7}+\frac {2 b^6 (d+e x)^{19/2}}{19 e^7}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 291, normalized size = 1.57 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (138567 a^6 e^6+92378 a^5 b e^5 (-2 d+7 e x)+20995 a^4 b^2 e^4 \left (8 d^2-28 d e x+63 e^2 x^2\right )+6460 a^3 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 a^2 b^4 e^2 \left (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\right )+38 a b^5 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+b^6 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{969969 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(138567*a^6*e^6 + 92378*a^5*b*e^5*(-2*d + 7*e*x) + 20995*a^4*b^2*e^4*(8*d^2 - 28*d*e*x + 63
*e^2*x^2) + 6460*a^3*b^3*e^3*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 323*a^2*b^4*e^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) + 38*a*b^5*e*(-256*d^5 + 896*d^4*e*x - 2016*d
^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006*d*e^4*x^4 + 9009*e^5*x^5) + b^6*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x
^2 - 14784*d^3*e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6)))/(969969*e^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(456\) vs. \(2(159)=318\).
time = 0.66, size = 457, normalized size = 2.47 Too large to display

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)^3,x,method=_RETURNVERBOSE)

[Out]

2/e^7*(1/19*b^6*(e*x+d)^(19/2)+3/17*(2*a*b*e-2*b^2*d)*b^4*(e*x+d)^(17/2)+1/15*((a^2*e^2-2*a*b*d*e+b^2*d^2)*b^4
+2*(2*a*b*e-2*b^2*d)^2*b^2+b^2*(2*(a^2*e^2-2*a*b*d*e+b^2*d^2)*b^2+(2*a*b*e-2*b^2*d)^2))*(e*x+d)^(15/2)+1/13*(4
*(a^2*e^2-2*a*b*d*e+b^2*d^2)*(2*a*b*e-2*b^2*d)*b^2+(2*a*b*e-2*b^2*d)*(2*(a^2*e^2-2*a*b*d*e+b^2*d^2)*b^2+(2*a*b
*e-2*b^2*d)^2))*(e*x+d)^(13/2)+1/11*((a^2*e^2-2*a*b*d*e+b^2*d^2)*(2*(a^2*e^2-2*a*b*d*e+b^2*d^2)*b^2+(2*a*b*e-2
*b^2*d)^2)+2*(2*a*b*e-2*b^2*d)^2*(a^2*e^2-2*a*b*d*e+b^2*d^2)+b^2*(a^2*e^2-2*a*b*d*e+b^2*d^2)^2)*(e*x+d)^(11/2)
+1/3*(a^2*e^2-2*a*b*d*e+b^2*d^2)^2*(2*a*b*e-2*b^2*d)*(e*x+d)^(9/2)+1/7*(a^2*e^2-2*a*b*d*e+b^2*d^2)^3*(e*x+d)^(
7/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (165) = 330\).
time = 0.28, size = 347, normalized size = 1.88 \begin {gather*} \frac {2}{969969} \, {\left (51051 \, {\left (x e + d\right )}^{\frac {19}{2}} b^{6} - 342342 \, {\left (b^{6} d - a b^{5} e\right )} {\left (x e + d\right )}^{\frac {17}{2}} + 969969 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (x e + d\right )}^{\frac {15}{2}} - 1492260 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {13}{2}} + 1322685 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {11}{2}} - 646646 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 138567 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-7\right )} \end {gather*}

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)^3,x, algorithm="maxima")

[Out]

2/969969*(51051*(x*e + d)^(19/2)*b^6 - 342342*(b^6*d - a*b^5*e)*(x*e + d)^(17/2) + 969969*(b^6*d^2 - 2*a*b^5*d
*e + a^2*b^4*e^2)*(x*e + d)^(15/2) - 1492260*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(x*e +
d)^(13/2) + 1322685*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(x*e + d)^(1
1/2) - 646646*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5
)*(x*e + d)^(9/2) + 138567*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2
*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*(x*e + d)^(7/2))*e^(-7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (165) = 330\).
time = 2.58, size = 599, normalized size = 3.24 \begin {gather*} \frac {2}{969969} \, {\left (1024 \, b^{6} d^{9} + {\left (51051 \, b^{6} x^{9} + 342342 \, a b^{5} x^{8} + 969969 \, a^{2} b^{4} x^{7} + 1492260 \, a^{3} b^{3} x^{6} + 1322685 \, a^{4} b^{2} x^{5} + 646646 \, a^{5} b x^{4} + 138567 \, a^{6} x^{3}\right )} e^{9} + {\left (117117 \, b^{6} d x^{8} + 798798 \, a b^{5} d x^{7} + 2313003 \, a^{2} b^{4} d x^{6} + 3662820 \, a^{3} b^{3} d x^{5} + 3380195 \, a^{4} b^{2} d x^{4} + 1755182 \, a^{5} b d x^{3} + 415701 \, a^{6} d x^{2}\right )} e^{8} + {\left (69069 \, b^{6} d^{2} x^{7} + 482790 \, a b^{5} d^{2} x^{6} + 1444779 \, a^{2} b^{4} d^{2} x^{5} + 2396660 \, a^{3} b^{3} d^{2} x^{4} + 2372435 \, a^{4} b^{2} d^{2} x^{3} + 1385670 \, a^{5} b d^{2} x^{2} + 415701 \, a^{6} d^{2} x\right )} e^{7} + {\left (231 \, b^{6} d^{3} x^{6} + 2394 \, a b^{5} d^{3} x^{5} + 11305 \, a^{2} b^{4} d^{3} x^{4} + 32300 \, a^{3} b^{3} d^{3} x^{3} + 62985 \, a^{4} b^{2} d^{3} x^{2} + 92378 \, a^{5} b d^{3} x + 138567 \, a^{6} d^{3}\right )} e^{6} - 4 \, {\left (63 \, b^{6} d^{4} x^{5} + 665 \, a b^{5} d^{4} x^{4} + 3230 \, a^{2} b^{4} d^{4} x^{3} + 9690 \, a^{3} b^{3} d^{4} x^{2} + 20995 \, a^{4} b^{2} d^{4} x + 46189 \, a^{5} b d^{4}\right )} e^{5} + 8 \, {\left (35 \, b^{6} d^{5} x^{4} + 380 \, a b^{5} d^{5} x^{3} + 1938 \, a^{2} b^{4} d^{5} x^{2} + 6460 \, a^{3} b^{3} d^{5} x + 20995 \, a^{4} b^{2} d^{5}\right )} e^{4} - 64 \, {\left (5 \, b^{6} d^{6} x^{3} + 57 \, a b^{5} d^{6} x^{2} + 323 \, a^{2} b^{4} d^{6} x + 1615 \, a^{3} b^{3} d^{6}\right )} e^{3} + 128 \, {\left (3 \, b^{6} d^{7} x^{2} + 38 \, a b^{5} d^{7} x + 323 \, a^{2} b^{4} d^{7}\right )} e^{2} - 512 \, {\left (b^{6} d^{8} x + 19 \, a b^{5} d^{8}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}

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)^3,x, algorithm="fricas")

[Out]

2/969969*(1024*b^6*d^9 + (51051*b^6*x^9 + 342342*a*b^5*x^8 + 969969*a^2*b^4*x^7 + 1492260*a^3*b^3*x^6 + 132268
5*a^4*b^2*x^5 + 646646*a^5*b*x^4 + 138567*a^6*x^3)*e^9 + (117117*b^6*d*x^8 + 798798*a*b^5*d*x^7 + 2313003*a^2*
b^4*d*x^6 + 3662820*a^3*b^3*d*x^5 + 3380195*a^4*b^2*d*x^4 + 1755182*a^5*b*d*x^3 + 415701*a^6*d*x^2)*e^8 + (690
69*b^6*d^2*x^7 + 482790*a*b^5*d^2*x^6 + 1444779*a^2*b^4*d^2*x^5 + 2396660*a^3*b^3*d^2*x^4 + 2372435*a^4*b^2*d^
2*x^3 + 1385670*a^5*b*d^2*x^2 + 415701*a^6*d^2*x)*e^7 + (231*b^6*d^3*x^6 + 2394*a*b^5*d^3*x^5 + 11305*a^2*b^4*
d^3*x^4 + 32300*a^3*b^3*d^3*x^3 + 62985*a^4*b^2*d^3*x^2 + 92378*a^5*b*d^3*x + 138567*a^6*d^3)*e^6 - 4*(63*b^6*
d^4*x^5 + 665*a*b^5*d^4*x^4 + 3230*a^2*b^4*d^4*x^3 + 9690*a^3*b^3*d^4*x^2 + 20995*a^4*b^2*d^4*x + 46189*a^5*b*
d^4)*e^5 + 8*(35*b^6*d^5*x^4 + 380*a*b^5*d^5*x^3 + 1938*a^2*b^4*d^5*x^2 + 6460*a^3*b^3*d^5*x + 20995*a^4*b^2*d
^5)*e^4 - 64*(5*b^6*d^6*x^3 + 57*a*b^5*d^6*x^2 + 323*a^2*b^4*d^6*x + 1615*a^3*b^3*d^6)*e^3 + 128*(3*b^6*d^7*x^
2 + 38*a*b^5*d^7*x + 323*a^2*b^4*d^7)*e^2 - 512*(b^6*d^8*x + 19*a*b^5*d^8)*e)*sqrt(x*e + d)*e^(-7)

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Sympy [A]
time = 29.17, size = 1671, normalized size = 9.03 \begin {gather*} \text {Too large to display} \end {gather*}

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)**3,x)

[Out]

a**6*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**6*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**6*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
12*a**5*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 24*a**5*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 + 12*a**5*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 + 30*a**4*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 + 60*a**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 + 30*a**4*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 + 40*a**3*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
+ 80*a**3*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 + 40*a**3*b**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 1
0*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
+ 30*a**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 + 60*a**2*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 + 30*a**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
+ 12*a*b**5*d**2*(-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**6 + 24*a*b**5*d*(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**6 + 12*a*b**5*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*
(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*
d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6 + 2*b**6*d**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**7 + 4*b**6*d*(-d**7*(d + e*x)**(3/2)/3
+ 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2
)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 2*b**6*(d**8*(d
+ e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(
d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 +
(d + e*x)**(19/2)/19)/e**7

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2171 vs. \(2 (165) = 330\).
time = 0.90, size = 2171, normalized size = 11.74 \begin {gather*} \text {Too large to display} \end {gather*}

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)^3,x, algorithm="giac")

[Out]

2/4849845*(9699690*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d^3*e^(-1) + 4849845*(3*(x*e + d)^(5/2) - 10*(x
*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*d^3*e^(-2) + 2771340*(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^3*d^3*e^(-3) + 230945*(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^4*d^3*e^(-4) +
41990*(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^5*d^3*e^(-5) + 1615*(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^6*d^3*e^(-6) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x
*e + d)*d^2)*a^5*b*d^2*e^(-1) + 6235515*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 3
5*sqrt(x*e + d)*d^3)*a^4*b^2*d^2*e^(-2) + 923780*(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^3*d^2*e^(-3) + 314925*(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*s
qrt(x*e + d)*d^5)*a^2*b^4*d^2*e^(-4) + 29070*(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^5*d^2*e^(-5) + 2261*(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 + 15015*(x*e + d)^(3/2)*d^6
- 6435*sqrt(x*e + d)*d^7)*b^6*d^2*e^(-6) + 4849845*sqrt(x*e + d)*a^6*d^3 + 4849845*((x*e + d)^(3/2) - 3*sqrt(x
*e + d)*d)*a^6*d^2 + 2494206*(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^5*b*d*e^(-1) + 692835*(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^4*b^2*d*e^(-2) + 419900*(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^3*b^3*d*e^(-3) + 72675*(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^2*b^4*d*e^
(-4) + 13566*(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 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e +
d)*d^7)*a*b^5*d*e^(-5) + 133*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 -
556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2
)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b^6*d*e^(-6) + 969969*(3*(x*e + d)^(5/2) - 10*(
x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^6*d + 92378*(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^5*b*e^(-1) + 104975*(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*s
qrt(x*e + d)*d^5)*a^4*b^2*e^(-2) + 32300*(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^3*b^3*e^(-3) + 11305*(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 + 15015*(x*e + d)^(3/2)*d^6 - 643
5*sqrt(x*e + d)*d^7)*a^2*b^4*e^(-4) + 266*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)
^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(
x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*a*b^5*e^(-5) + 21*(12155*(x*e + d)
^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*
e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 +
692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*b^6*e^(-6) + 138567*(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^6)*e^(-1)

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Mupad [B]
time = 0.06, size = 162, normalized size = 0.88 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^7}+\frac {2\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}+\frac {2\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{15/2}}{e^7}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{9/2}}{3\,e^7} \end {gather*}

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)^3,x)

[Out]

(2*b^6*(d + e*x)^(19/2))/(19*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(17/2))/(17*e^7) + (2*(a*e - b*d)^6*(d
+ e*x)^(7/2))/(7*e^7) + (30*b^2*(a*e - b*d)^4*(d + e*x)^(11/2))/(11*e^7) + (40*b^3*(a*e - b*d)^3*(d + e*x)^(13
/2))/(13*e^7) + (2*b^4*(a*e - b*d)^2*(d + e*x)^(15/2))/e^7 + (4*b*(a*e - b*d)^5*(d + e*x)^(9/2))/(3*e^7)

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