\(\int (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1629]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (b d-a e)^4 (d+e x)^{9/2}}{9 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{11/2}}{11 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{13/2}}{13 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{15/2}}{15 e^5}+\frac {2 b^4 (d+e x)^{17/2}}{17 e^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {8 b^3 (d+e x)^{15/2} (b d-a e)}{15 e^5}+\frac {12 b^2 (d+e x)^{13/2} (b d-a e)^2}{13 e^5}-\frac {8 b (d+e x)^{11/2} (b d-a e)^3}{11 e^5}+\frac {2 (d+e x)^{9/2} (b d-a e)^4}{9 e^5}+\frac {2 b^4 (d+e x)^{17/2}}{17 e^5} \]

[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 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*} \text {integral}& = \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] (verified)

Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{9/2} \left (12155 a^4 e^4+4420 a^3 b e^3 (-2 d+9 e x)+510 a^2 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+68 a b^3 e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+b^4 \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )}{109395 e^5} \]

[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*a^4*e^4 + 4420*a^3*b*e^3*(-2*d + 9*e*x) + 510*a^2*b^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2
*x^2) + 68*a*b^3*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + b^4*(128*d^4 - 576*d^3*e*x + 1584*d^
2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4)))/(109395*e^5)

Maple [A] (verified)

Time = 2.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (\left (\frac {9}{17} b^{4} x^{4}+\frac {12}{5} a \,b^{3} x^{3}+\frac {54}{13} a^{2} b^{2} x^{2}+\frac {36}{11} a^{3} b x +a^{4}\right ) e^{4}-\frac {8 b \left (\frac {33}{85} b^{3} x^{3}+\frac {99}{65} a \,b^{2} x^{2}+\frac {27}{13} a^{2} b x +a^{3}\right ) d \,e^{3}}{11}+\frac {48 b^{2} \left (\frac {33}{85} b^{2} x^{2}+\frac {6}{5} a b x +a^{2}\right ) d^{2} e^{2}}{143}-\frac {64 b^{3} \left (\frac {9 b x}{17}+a \right ) d^{3} e}{715}+\frac {128 b^{4} d^{4}}{12155}\right )}{9 e^{5}}\)	\(143\)
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\)	\(167\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{5}}\)	\(167\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (6435 b^{4} x^{4} e^{4}+29172 x^{3} a \,b^{3} e^{4}-3432 x^{3} b^{4} d \,e^{3}+50490 x^{2} a^{2} b^{2} e^{4}-13464 x^{2} a \,b^{3} d \,e^{3}+1584 x^{2} b^{4} d^{2} e^{2}+39780 x \,a^{3} b \,e^{4}-18360 x \,a^{2} b^{2} d \,e^{3}+4896 x a \,b^{3} d^{2} e^{2}-576 x \,b^{4} d^{3} e +12155 e^{4} a^{4}-8840 b \,e^{3} d \,a^{3}+4080 b^{2} e^{2} d^{2} a^{2}-1088 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{109395 e^{5}}\)	\(186\)
trager	\(\frac {2 \left (6435 b^{4} e^{8} x^{8}+29172 a \,b^{3} e^{8} x^{7}+22308 b^{4} d \,e^{7} x^{7}+50490 a^{2} b^{2} e^{8} x^{6}+103224 a \,b^{3} d \,e^{7} x^{6}+26466 b^{4} d^{2} e^{6} x^{6}+39780 a^{3} b \,e^{8} x^{5}+183600 a^{2} b^{2} d \,e^{7} x^{5}+126072 a \,b^{3} d^{2} e^{6} x^{5}+10908 b^{4} d^{3} e^{5} x^{5}+12155 a^{4} e^{8} x^{4}+150280 a^{3} b d \,e^{7} x^{4}+233580 a^{2} b^{2} d^{2} e^{6} x^{4}+54400 a \,b^{3} d^{3} e^{5} x^{4}+35 b^{4} d^{4} e^{4} x^{4}+48620 a^{4} d \,e^{7} x^{3}+203320 a^{3} b \,d^{2} e^{6} x^{3}+108120 a^{2} b^{2} d^{3} e^{5} x^{3}+340 a \,b^{3} d^{4} e^{4} x^{3}-40 b^{4} d^{5} e^{3} x^{3}+72930 a^{4} d^{2} e^{6} x^{2}+106080 a^{3} b \,d^{3} e^{5} x^{2}+1530 a^{2} b^{2} d^{4} e^{4} x^{2}-408 a \,b^{3} d^{5} e^{3} x^{2}+48 b^{4} d^{6} e^{2} x^{2}+48620 a^{4} d^{3} e^{5} x +4420 a^{3} b \,d^{4} e^{4} x -2040 a^{2} b^{2} d^{5} e^{3} x +544 a \,b^{3} d^{6} e^{2} x -64 b^{4} d^{7} e x +12155 a^{4} d^{4} e^{4}-8840 a^{3} b \,d^{5} e^{3}+4080 a^{2} b^{2} d^{6} e^{2}-1088 a \,b^{3} d^{7} e +128 b^{4} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\)	\(482\)
risch	\(\frac {2 \left (6435 b^{4} e^{8} x^{8}+29172 a \,b^{3} e^{8} x^{7}+22308 b^{4} d \,e^{7} x^{7}+50490 a^{2} b^{2} e^{8} x^{6}+103224 a \,b^{3} d \,e^{7} x^{6}+26466 b^{4} d^{2} e^{6} x^{6}+39780 a^{3} b \,e^{8} x^{5}+183600 a^{2} b^{2} d \,e^{7} x^{5}+126072 a \,b^{3} d^{2} e^{6} x^{5}+10908 b^{4} d^{3} e^{5} x^{5}+12155 a^{4} e^{8} x^{4}+150280 a^{3} b d \,e^{7} x^{4}+233580 a^{2} b^{2} d^{2} e^{6} x^{4}+54400 a \,b^{3} d^{3} e^{5} x^{4}+35 b^{4} d^{4} e^{4} x^{4}+48620 a^{4} d \,e^{7} x^{3}+203320 a^{3} b \,d^{2} e^{6} x^{3}+108120 a^{2} b^{2} d^{3} e^{5} x^{3}+340 a \,b^{3} d^{4} e^{4} x^{3}-40 b^{4} d^{5} e^{3} x^{3}+72930 a^{4} d^{2} e^{6} x^{2}+106080 a^{3} b \,d^{3} e^{5} x^{2}+1530 a^{2} b^{2} d^{4} e^{4} x^{2}-408 a \,b^{3} d^{5} e^{3} x^{2}+48 b^{4} d^{6} e^{2} x^{2}+48620 a^{4} d^{3} e^{5} x +4420 a^{3} b \,d^{4} e^{4} x -2040 a^{2} b^{2} d^{5} e^{3} x +544 a \,b^{3} d^{6} e^{2} x -64 b^{4} d^{7} e x +12155 a^{4} d^{4} e^{4}-8840 a^{3} b \,d^{5} e^{3}+4080 a^{2} b^{2} d^{6} e^{2}-1088 a \,b^{3} d^{7} e +128 b^{4} d^{8}\right ) \sqrt {e x +d}}{109395 e^{5}}\)	\(482\)

[In]

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

[Out]

2/9*(e*x+d)^(9/2)*((9/17*b^4*x^4+12/5*a*b^3*x^3+54/13*a^2*b^2*x^2+36/11*a^3*b*x+a^4)*e^4-8/11*b*(33/85*b^3*x^3
+99/65*a*b^2*x^2+27/13*a^2*b*x+a^3)*d*e^3+48/143*b^2*(33/85*b^2*x^2+6/5*a*b*x+a^2)*d^2*e^2-64/715*b^3*(9/17*b*
x+a)*d^3*e+128/12155*b^4*d^4)/e^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (109) = 218\).

Time = 0.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.44 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (6435 \, b^{4} e^{8} x^{8} + 128 \, b^{4} d^{8} - 1088 \, a b^{3} d^{7} e + 4080 \, a^{2} b^{2} d^{6} e^{2} - 8840 \, a^{3} b d^{5} e^{3} + 12155 \, a^{4} d^{4} e^{4} + 1716 \, {\left (13 \, b^{4} d e^{7} + 17 \, a b^{3} e^{8}\right )} x^{7} + 66 \, {\left (401 \, b^{4} d^{2} e^{6} + 1564 \, a b^{3} d e^{7} + 765 \, a^{2} b^{2} e^{8}\right )} x^{6} + 36 \, {\left (303 \, b^{4} d^{3} e^{5} + 3502 \, a b^{3} d^{2} e^{6} + 5100 \, a^{2} b^{2} d e^{7} + 1105 \, a^{3} b e^{8}\right )} x^{5} + 5 \, {\left (7 \, b^{4} d^{4} e^{4} + 10880 \, a b^{3} d^{3} e^{5} + 46716 \, a^{2} b^{2} d^{2} e^{6} + 30056 \, a^{3} b d e^{7} + 2431 \, a^{4} e^{8}\right )} x^{4} - 20 \, {\left (2 \, b^{4} d^{5} e^{3} - 17 \, a b^{3} d^{4} e^{4} - 5406 \, a^{2} b^{2} d^{3} e^{5} - 10166 \, a^{3} b d^{2} e^{6} - 2431 \, a^{4} d e^{7}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{6} e^{2} - 68 \, a b^{3} d^{5} e^{3} + 255 \, a^{2} b^{2} d^{4} e^{4} + 17680 \, a^{3} b d^{3} e^{5} + 12155 \, a^{4} d^{2} e^{6}\right )} x^{2} - 4 \, {\left (16 \, b^{4} d^{7} e - 136 \, a b^{3} d^{6} e^{2} + 510 \, a^{2} b^{2} d^{5} e^{3} - 1105 \, a^{3} b d^{4} e^{4} - 12155 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{5}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (119) = 238\).

Time = 0.75 (sec) , antiderivative size = 903, normalized size of antiderivative = 7.00 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 a^{4} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{4} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{4} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{4} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{4} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {16 a^{3} b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {8 a^{3} b d^{4} x \sqrt {d + e x}}{99 e} + \frac {64 a^{3} b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {368 a^{3} b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {272 a^{3} b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {8 a^{3} b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {32 a^{2} b^{2} d^{6} \sqrt {d + e x}}{429 e^{3}} - \frac {16 a^{2} b^{2} d^{5} x \sqrt {d + e x}}{429 e^{2}} + \frac {4 a^{2} b^{2} d^{4} x^{2} \sqrt {d + e x}}{143 e} + \frac {848 a^{2} b^{2} d^{3} x^{3} \sqrt {d + e x}}{429} + \frac {1832 a^{2} b^{2} d^{2} e x^{4} \sqrt {d + e x}}{429} + \frac {480 a^{2} b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {12 a^{2} b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {128 a b^{3} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {64 a b^{3} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {16 a b^{3} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {8 a b^{3} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {1280 a b^{3} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {1648 a b^{3} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {368 a b^{3} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {8 a b^{3} e^{3} x^{7} \sqrt {d + e x}}{15} + \frac {256 b^{4} d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {128 b^{4} d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {32 b^{4} d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {16 b^{4} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {14 b^{4} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 b^{4} d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {1604 b^{4} d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {104 b^{4} d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {2 b^{4} e^{3} x^{8} \sqrt {d + e x}}{17} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{4} - 29172 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 50490 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 39780 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 12155 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{109395 \, e^{5}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1661 vs. \(2 (109) = 218\).

Time = 0.30 (sec) , antiderivative size = 1661, normalized size of antiderivative = 12.88 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/765765*(765765*sqrt(e*x + d)*a^4*d^4 + 1021020*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*d^3 + 1021020*((e*x
+ d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*b*d^4/e + 306306*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x
+ d)*d^2)*a^4*d^2 + 306306*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b^2*d^4/e^2 +
816816*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b*d^3/e + 87516*(5*(e*x + d)^(7/
2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^4*d + 87516*(5*(e*x + d)^(7/2) -
21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*b^3*d^4/e^3 + 525096*(5*(e*x + d)^(7/2
) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*b^2*d^3/e^2 + 525096*(5*(e*x + d
)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^3*b*d^2/e + 2431*(35*(e*x +
d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*
a^4 + 2431*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 3
15*sqrt(e*x + d)*d^4)*b^4*d^4/e^4 + 38896*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^
2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b^3*d^3/e^3 + 87516*(35*(e*x + d)^(9/2) - 180*(e*x + d)
^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^2*b^2*d^2/e^2 + 38896*
(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x
+ d)*d^4)*a^3*b*d/e + 4420*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x
+ d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^4*d^3/e^4 + 26520*(63*(e*x + d)^(11/2) -
385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*s
qrt(e*x + d)*d^5)*a*b^3*d^2/e^3 + 26520*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2
- 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*a^2*b^2*d/e^2 + 4420*(63*(e*x
+ d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2
)*d^4 - 693*sqrt(e*x + d)*d^5)*a^3*b/e + 1530*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)
^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x +
d)*d^6)*b^4*d^2/e^4 + 4080*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(
e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*a*b^3*d/e^3
+ 1530*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3
+ 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*a^2*b^2/e^2 + 476*(429*(e*x +
d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d^3 + 32175*(e*x + d)
^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d^7)*b^4*d/e^4 + 476*(
429*(e*x + d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)*d^3 + 3217
5*(e*x + d)^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d^7)*a*b^3/
e^3 + 7*(6435*(e*x + d)^(17/2) - 58344*(e*x + d)^(15/2)*d + 235620*(e*x + d)^(13/2)*d^2 - 556920*(e*x + d)^(11
/2)*d^3 + 850850*(e*x + d)^(9/2)*d^4 - 875160*(e*x + d)^(7/2)*d^5 + 612612*(e*x + d)^(5/2)*d^6 - 291720*(e*x +
d)^(3/2)*d^7 + 109395*sqrt(e*x + d)*d^8)*b^4/e^4)/e

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^4\,{\left (d+e\,x\right )}^{17/2}}{17\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5} \]

[In]

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

[Out]

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