3.17.30 \(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1630]

Optimal. Leaf size=129 \[ \frac {2 (b d-a e)^4 (d+e x)^{7/2}}{7 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{9/2}}{9 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{11/2}}{11 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{13/2}}{13 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5} \]

[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 = 0.03, 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, 45} \begin {gather*} -\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} \end {gather*}

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 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 )^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 = 0.09, size = 154, normalized size = 1.19 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (6435 a^4 e^4+2860 a^3 b e^3 (-2 d+7 e x)+390 a^2 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+60 a b^3 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^4 \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 )\right )}{45045 e^5} \end {gather*}

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*a^4*e^4 + 2860*a^3*b*e^3*(-2*d + 7*e*x) + 390*a^2*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*
x^2) + 60*a*b^3*e*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^4*(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)))/(45045*e^5)

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Maple [A]
time = 0.64, size = 167, normalized size = 1.29

method	result	size
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 \left (2 a b e -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(167\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 \left (2 a b e -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(167\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 b^{4} x^{4} e^{4}+13860 a \,b^{3} e^{4} x^{3}-1848 b^{4} d \,e^{3} x^{3}+24570 a^{2} b^{2} e^{4} x^{2}-7560 a \,b^{3} d \,e^{3} x^{2}+1008 b^{4} d^{2} e^{2} x^{2}+20020 a^{3} b \,e^{4} x -10920 a^{2} b^{2} d \,e^{3} x +3360 a \,b^{3} d^{2} e^{2} x -448 b^{4} d^{3} e x +6435 e^{4} a^{4}-5720 a^{3} b d \,e^{3}+3120 a^{2} b^{2} d^{2} e^{2}-960 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{45045 e^{5}}\)	\(186\)
trager	\(\frac {2 \left (3003 b^{4} e^{7} x^{7}+13860 a \,b^{3} e^{7} x^{6}+7161 b^{4} d \,e^{6} x^{6}+24570 a^{2} b^{2} e^{7} x^{5}+34020 a \,b^{3} d \,e^{6} x^{5}+4473 b^{4} d^{2} e^{5} x^{5}+20020 a^{3} b \,e^{7} x^{4}+62790 a^{2} b^{2} d \,e^{6} x^{4}+22260 a \,b^{3} d^{2} e^{5} x^{4}+35 b^{4} d^{3} e^{4} x^{4}+6435 e^{7} a^{4} x^{3}+54340 a^{3} b d \,e^{6} x^{3}+44070 a^{2} b^{2} d^{2} e^{5} x^{3}+300 a \,b^{3} d^{3} e^{4} x^{3}-40 b^{4} d^{4} e^{3} x^{3}+19305 a^{4} d \,e^{6} x^{2}+42900 a^{3} b \,d^{2} e^{5} x^{2}+1170 a^{2} b^{2} d^{3} e^{4} x^{2}-360 a \,b^{3} d^{4} e^{3} x^{2}+48 b^{4} d^{5} e^{2} x^{2}+19305 a^{4} d^{2} e^{5} x +2860 a^{3} b \,d^{3} e^{4} x -1560 a^{2} b^{2} d^{4} e^{3} x +480 a \,b^{3} d^{5} e^{2} x -64 b^{4} d^{6} e x +6435 a^{4} d^{3} e^{4}-5720 a^{3} b \,d^{4} e^{3}+3120 a^{2} b^{2} d^{5} e^{2}-960 a \,b^{3} d^{6} e +128 b^{4} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(407\)
risch	\(\frac {2 \left (3003 b^{4} e^{7} x^{7}+13860 a \,b^{3} e^{7} x^{6}+7161 b^{4} d \,e^{6} x^{6}+24570 a^{2} b^{2} e^{7} x^{5}+34020 a \,b^{3} d \,e^{6} x^{5}+4473 b^{4} d^{2} e^{5} x^{5}+20020 a^{3} b \,e^{7} x^{4}+62790 a^{2} b^{2} d \,e^{6} x^{4}+22260 a \,b^{3} d^{2} e^{5} x^{4}+35 b^{4} d^{3} e^{4} x^{4}+6435 e^{7} a^{4} x^{3}+54340 a^{3} b d \,e^{6} x^{3}+44070 a^{2} b^{2} d^{2} e^{5} x^{3}+300 a \,b^{3} d^{3} e^{4} x^{3}-40 b^{4} d^{4} e^{3} x^{3}+19305 a^{4} d \,e^{6} x^{2}+42900 a^{3} b \,d^{2} e^{5} x^{2}+1170 a^{2} b^{2} d^{3} e^{4} x^{2}-360 a \,b^{3} d^{4} e^{3} x^{2}+48 b^{4} d^{5} e^{2} x^{2}+19305 a^{4} d^{2} e^{5} x +2860 a^{3} b \,d^{3} e^{4} x -1560 a^{2} b^{2} d^{4} e^{3} x +480 a \,b^{3} d^{5} e^{2} x -64 b^{4} d^{6} e x +6435 a^{4} d^{3} e^{4}-5720 a^{3} b \,d^{4} e^{3}+3120 a^{2} b^{2} d^{5} e^{2}-960 a \,b^{3} d^{6} e +128 b^{4} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(407\)

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,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/15*b^4*(e*x+d)^(15/2)+2/13*(2*a*b*e-2*b^2*d)*b^2*(e*x+d)^(13/2)+1/11*(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)^(11/2)+2/9*(a^2*e^2-2*a*b*d*e+b^2*d^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)^2*(e*x+d)^(7/2))

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Maxima [A]
time = 0.30, size = 183, normalized size = 1.42 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, {\left (x e + d\right )}^{\frac {15}{2}} b^{4} - 13860 \, {\left (b^{4} d - a b^{3} e\right )} {\left (x e + 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 (x e + 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 (x e + 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 (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-5\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)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(x*e + d)^(15/2)*b^4 - 13860*(b^4*d - a*b^3*e)*(x*e + d)^(13/2) + 24570*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(x*e + 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)*(x*e + 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)*(x*e + d)^(7/2))*e^(-5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (113) = 226\).
time = 3.03, size = 353, normalized size = 2.74 \begin {gather*} \frac {2}{45045} \, {\left (128 \, b^{4} d^{7} + {\left (3003 \, b^{4} x^{7} + 13860 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{5} + 20020 \, a^{3} b x^{4} + 6435 \, a^{4} x^{3}\right )} e^{7} + {\left (7161 \, b^{4} d x^{6} + 34020 \, a b^{3} d x^{5} + 62790 \, a^{2} b^{2} d x^{4} + 54340 \, a^{3} b d x^{3} + 19305 \, a^{4} d x^{2}\right )} e^{6} + 3 \, {\left (1491 \, b^{4} d^{2} x^{5} + 7420 \, a b^{3} d^{2} x^{4} + 14690 \, a^{2} b^{2} d^{2} x^{3} + 14300 \, a^{3} b d^{2} x^{2} + 6435 \, a^{4} d^{2} x\right )} e^{5} + 5 \, {\left (7 \, b^{4} d^{3} x^{4} + 60 \, a b^{3} d^{3} x^{3} + 234 \, a^{2} b^{2} d^{3} x^{2} + 572 \, a^{3} b d^{3} x + 1287 \, a^{4} d^{3}\right )} e^{4} - 40 \, {\left (b^{4} d^{4} x^{3} + 9 \, a b^{3} d^{4} x^{2} + 39 \, a^{2} b^{2} d^{4} x + 143 \, a^{3} b d^{4}\right )} e^{3} + 48 \, {\left (b^{4} d^{5} x^{2} + 10 \, a b^{3} d^{5} x + 65 \, a^{2} b^{2} d^{5}\right )} e^{2} - 64 \, {\left (b^{4} d^{6} x + 15 \, a b^{3} d^{6}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\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)^2,x, algorithm="fricas")

[Out]

2/45045*(128*b^4*d^7 + (3003*b^4*x^7 + 13860*a*b^3*x^6 + 24570*a^2*b^2*x^5 + 20020*a^3*b*x^4 + 6435*a^4*x^3)*e
^7 + (7161*b^4*d*x^6 + 34020*a*b^3*d*x^5 + 62790*a^2*b^2*d*x^4 + 54340*a^3*b*d*x^3 + 19305*a^4*d*x^2)*e^6 + 3*
(1491*b^4*d^2*x^5 + 7420*a*b^3*d^2*x^4 + 14690*a^2*b^2*d^2*x^3 + 14300*a^3*b*d^2*x^2 + 6435*a^4*d^2*x)*e^5 + 5
*(7*b^4*d^3*x^4 + 60*a*b^3*d^3*x^3 + 234*a^2*b^2*d^3*x^2 + 572*a^3*b*d^3*x + 1287*a^4*d^3)*e^4 - 40*(b^4*d^4*x
^3 + 9*a*b^3*d^4*x^2 + 39*a^2*b^2*d^4*x + 143*a^3*b*d^4)*e^3 + 48*(b^4*d^5*x^2 + 10*a*b^3*d^5*x + 65*a^2*b^2*d
^5)*e^2 - 64*(b^4*d^6*x + 15*a*b^3*d^6)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [A]
time = 18.65, size = 960, normalized size = 7.44 \begin {gather*} a^{4} d^{2} \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {4 a^{4} d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {2 a^{4} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e} + \frac {8 a^{3} b d^{2} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {16 a^{3} b d \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {8 a^{3} b \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{2}} + \frac {12 a^{2} b^{2} d^{2} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {24 a^{2} b^{2} d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {12 a^{2} b^{2} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{3}} + \frac {8 a b^{3} d^{2} \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{4}} + \frac {16 a b^{3} d \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{4}} + \frac {8 a b^{3} \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{4}} + \frac {2 b^{4} d^{2} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {4 b^{4} d \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} + \frac {2 b^{4} \left (\frac {d^{6} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {6 d^{5} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {15 d^{4} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {20 d^{3} \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {15 d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {6 d \left (d + e x\right )^{\frac {13}{2}}}{13} + \frac {\left (d + e x\right )^{\frac {15}{2}}}{15}\right )}{e^{5}} \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)**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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1277 vs. \(2 (113) = 226\).
time = 0.85, size = 1277, normalized size = 9.90 \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)^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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Mupad [B]
time = 0.54, size = 112, normalized size = 0.87 \begin {gather*} \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} \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)^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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