3.17.31 \(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1631]

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

[Out]

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

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 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)^{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*}

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Mathematica [A]
time = 0.09, size = 154, normalized size = 1.19 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (3003 a^4 e^4+1716 a^3 b e^3 (-2 d+5 e x)+286 a^2 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+52 a b^3 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+b^4 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 e^5} \end {gather*}

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*a^4*e^4 + 1716*a^3*b*e^3*(-2*d + 5*e*x) + 286*a^2*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*
x^2) + 52*a*b^3*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + b^4*(128*d^4 - 320*d^3*e*x + 560*d^2*e
^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*e^5)

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Maple [A]
time = 0.66, size = 167, normalized size = 1.29

method	result	size
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 \left (2 a b e -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\)	\(167\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 \left (2 a b e -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{5}}\)	\(167\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 b^{4} x^{4} e^{4}+5460 a \,b^{3} e^{4} x^{3}-840 b^{4} d \,e^{3} x^{3}+10010 a^{2} b^{2} e^{4} x^{2}-3640 a \,b^{3} d \,e^{3} x^{2}+560 b^{4} d^{2} e^{2} x^{2}+8580 a^{3} b \,e^{4} x -5720 a^{2} b^{2} d \,e^{3} x +2080 a \,b^{3} d^{2} e^{2} x -320 b^{4} d^{3} e x +3003 e^{4} a^{4}-3432 a^{3} b d \,e^{3}+2288 a^{2} b^{2} d^{2} e^{2}-832 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{15015 e^{5}}\)	\(186\)
trager	\(\frac {2 \left (1155 b^{4} e^{6} x^{6}+5460 a \,b^{3} e^{6} x^{5}+1470 b^{4} d \,e^{5} x^{5}+10010 a^{2} b^{2} e^{6} x^{4}+7280 a \,b^{3} d \,e^{5} x^{4}+35 b^{4} d^{2} e^{4} x^{4}+8580 a^{3} b \,e^{6} x^{3}+14300 a^{2} b^{2} d \,e^{5} x^{3}+260 a \,b^{3} d^{2} e^{4} x^{3}-40 b^{4} d^{3} e^{3} x^{3}+3003 a^{4} e^{6} x^{2}+13728 a^{3} b d \,e^{5} x^{2}+858 a^{2} b^{2} d^{2} e^{4} x^{2}-312 a \,b^{3} d^{3} e^{3} x^{2}+48 b^{4} d^{4} e^{2} x^{2}+6006 a^{4} d \,e^{5} x +1716 a^{3} b \,d^{2} e^{4} x -1144 a^{2} b^{2} d^{3} e^{3} x +416 a \,b^{3} d^{4} e^{2} x -64 b^{4} d^{5} e x +3003 a^{4} d^{2} e^{4}-3432 a^{3} b \,d^{3} e^{3}+2288 a^{2} b^{2} d^{4} e^{2}-832 a \,b^{3} d^{5} e +128 b^{4} d^{6}\right ) \sqrt {e x +d}}{15015 e^{5}}\)	\(332\)
risch	\(\frac {2 \left (1155 b^{4} e^{6} x^{6}+5460 a \,b^{3} e^{6} x^{5}+1470 b^{4} d \,e^{5} x^{5}+10010 a^{2} b^{2} e^{6} x^{4}+7280 a \,b^{3} d \,e^{5} x^{4}+35 b^{4} d^{2} e^{4} x^{4}+8580 a^{3} b \,e^{6} x^{3}+14300 a^{2} b^{2} d \,e^{5} x^{3}+260 a \,b^{3} d^{2} e^{4} x^{3}-40 b^{4} d^{3} e^{3} x^{3}+3003 a^{4} e^{6} x^{2}+13728 a^{3} b d \,e^{5} x^{2}+858 a^{2} b^{2} d^{2} e^{4} x^{2}-312 a \,b^{3} d^{3} e^{3} x^{2}+48 b^{4} d^{4} e^{2} x^{2}+6006 a^{4} d \,e^{5} x +1716 a^{3} b \,d^{2} e^{4} x -1144 a^{2} b^{2} d^{3} e^{3} x +416 a \,b^{3} d^{4} e^{2} x -64 b^{4} d^{5} e x +3003 a^{4} d^{2} e^{4}-3432 a^{3} b \,d^{3} e^{3}+2288 a^{2} b^{2} d^{4} e^{2}-832 a \,b^{3} d^{5} e +128 b^{4} d^{6}\right ) \sqrt {e x +d}}{15015 e^{5}}\)	\(332\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 183, normalized size = 1.42 \begin {gather*} \frac {2}{15015} \, {\left (1155 \, {\left (x e + d\right )}^{\frac {13}{2}} b^{4} - 5460 \, {\left (b^{4} d - a b^{3} e\right )} {\left (x e + 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 (x e + 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 (x e + 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 (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-5\right )} \end {gather*}

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*(x*e + d)^(13/2)*b^4 - 5460*(b^4*d - a*b^3*e)*(x*e + d)^(11/2) + 10010*(b^4*d^2 - 2*a*b^3*d*e +
a^2*b^2*e^2)*(x*e + 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)*(x*e + 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)*(x*e + d)^(5/2))*e^(-5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (113) = 226\).
time = 2.85, size = 290, normalized size = 2.25 \begin {gather*} \frac {2}{15015} \, {\left (128 \, b^{4} d^{6} + {\left (1155 \, b^{4} x^{6} + 5460 \, a b^{3} x^{5} + 10010 \, a^{2} b^{2} x^{4} + 8580 \, a^{3} b x^{3} + 3003 \, a^{4} x^{2}\right )} e^{6} + 2 \, {\left (735 \, b^{4} d x^{5} + 3640 \, a b^{3} d x^{4} + 7150 \, a^{2} b^{2} d x^{3} + 6864 \, a^{3} b d x^{2} + 3003 \, a^{4} d x\right )} e^{5} + {\left (35 \, b^{4} d^{2} x^{4} + 260 \, a b^{3} d^{2} x^{3} + 858 \, a^{2} b^{2} d^{2} x^{2} + 1716 \, a^{3} b d^{2} x + 3003 \, a^{4} d^{2}\right )} e^{4} - 8 \, {\left (5 \, b^{4} d^{3} x^{3} + 39 \, a b^{3} d^{3} x^{2} + 143 \, a^{2} b^{2} d^{3} x + 429 \, a^{3} b d^{3}\right )} e^{3} + 16 \, {\left (3 \, b^{4} d^{4} x^{2} + 26 \, a b^{3} d^{4} x + 143 \, a^{2} b^{2} d^{4}\right )} e^{2} - 64 \, {\left (b^{4} d^{5} x + 13 \, a b^{3} d^{5}\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)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/15015*(128*b^4*d^6 + (1155*b^4*x^6 + 5460*a*b^3*x^5 + 10010*a^2*b^2*x^4 + 8580*a^3*b*x^3 + 3003*a^4*x^2)*e^6
+ 2*(735*b^4*d*x^5 + 3640*a*b^3*d*x^4 + 7150*a^2*b^2*d*x^3 + 6864*a^3*b*d*x^2 + 3003*a^4*d*x)*e^5 + (35*b^4*d
^2*x^4 + 260*a*b^3*d^2*x^3 + 858*a^2*b^2*d^2*x^2 + 1716*a^3*b*d^2*x + 3003*a^4*d^2)*e^4 - 8*(5*b^4*d^3*x^3 + 3
9*a*b^3*d^3*x^2 + 143*a^2*b^2*d^3*x + 429*a^3*b*d^3)*e^3 + 16*(3*b^4*d^4*x^2 + 26*a*b^3*d^4*x + 143*a^2*b^2*d^
4)*e^2 - 64*(b^4*d^5*x + 13*a*b^3*d^5)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [A]
time = 10.98, size = 559, normalized size = 4.33 \begin {gather*} a^{4} d \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 {2 a^{4} \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 {8 a^{3} b 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^{2}} + \frac {8 a^{3} b \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 {12 a^{2} b^{2} 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^{3}} + \frac {12 a^{2} b^{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^{3}} + \frac {8 a b^{3} 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^{4}} + \frac {8 a b^{3} \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 {2 b^{4} 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^{5}} + \frac {2 b^{4} \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}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (113) = 226\).
time = 0.92, size = 854, normalized size = 6.62 \begin {gather*} \frac {2}{45045} \, {\left (60060 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} b d^{2} e^{\left (-1\right )} + 18018 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} b^{2} d^{2} e^{\left (-2\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a b^{3} d^{2} e^{\left (-3\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} b^{4} d^{2} e^{\left (-4\right )} + 24024 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{3} b d e^{\left (-1\right )} + 15444 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} b^{2} d e^{\left (-2\right )} + 1144 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a b^{3} d e^{\left (-3\right )} + 130 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} b^{4} d e^{\left (-4\right )} + 45045 \, \sqrt {x e + d} a^{4} d^{2} + 30030 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{4} d + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{3} b e^{\left (-1\right )} + 858 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2} b^{2} e^{\left (-2\right )} + 260 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a b^{3} e^{\left (-3\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{4} e^{\left (-4\right )} + 3003 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{4}\right )} e^{\left (-1\right )} \end {gather*}

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*(60060*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d^2*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^2*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^2*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^2*e^(-4) + 24024*(3*(x*e + d)
^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b*d*e^(-1) + 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^2*b^2*d*e^(-2) + 1144*(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*b^3*d*e^(-3) +
130*(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*e^(-4) + 45045*sqrt(x*e + d)*a^4*d^2 + 30030*((x*e + d)^(3/2
) - 3*sqrt(x*e + d)*d)*a^4*d + 5148*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sq
rt(x*e + d)*d^3)*a^3*b*e^(-1) + 858*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 42
0*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*b^2*e^(-2) + 260*(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*e^(-3) + 15*(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*e^(-4) + 3003*(3*(
x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4)*e^(-1)

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Mupad [B]
time = 0.04, size = 112, normalized size = 0.87 \begin {gather*} \frac {2\,b^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {4\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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