\(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{5/2}} \, dx\) [1643]

Optimal result

Integrand size = 28, antiderivative size = 181 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=-\frac {2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac {12 b (b d-a e)^5}{e^7 \sqrt {d+e x}}+\frac {30 b^2 (b d-a e)^4 \sqrt {d+e x}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{3/2}}{3 e^7}+\frac {6 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{7/2}}{7 e^7}+\frac {2 b^6 (d+e x)^{9/2}}{9 e^7} \]

[Out]

-2/3*(-a*e+b*d)^6/e^7/(e*x+d)^(3/2)-40/3*b^3*(-a*e+b*d)^3*(e*x+d)^(3/2)/e^7+6*b^4*(-a*e+b*d)^2*(e*x+d)^(5/2)/e
^7-12/7*b^5*(-a*e+b*d)*(e*x+d)^(7/2)/e^7+2/9*b^6*(e*x+d)^(9/2)/e^7+12*b*(-a*e+b*d)^5/e^7/(e*x+d)^(1/2)+30*b^2*
(-a*e+b*d)^4*(e*x+d)^(1/2)/e^7

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 181, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=-\frac {12 b^5 (d+e x)^{7/2} (b d-a e)}{7 e^7}+\frac {6 b^4 (d+e x)^{5/2} (b d-a e)^2}{e^7}-\frac {40 b^3 (d+e x)^{3/2} (b d-a e)^3}{3 e^7}+\frac {30 b^2 \sqrt {d+e x} (b d-a e)^4}{e^7}+\frac {12 b (b d-a e)^5}{e^7 \sqrt {d+e x}}-\frac {2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac {2 b^6 (d+e x)^{9/2}}{9 e^7} \]

[In]

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

[Out]

(-2*(b*d - a*e)^6)/(3*e^7*(d + e*x)^(3/2)) + (12*b*(b*d - a*e)^5)/(e^7*Sqrt[d + e*x]) + (30*b^2*(b*d - a*e)^4*
Sqrt[d + e*x])/e^7 - (40*b^3*(b*d - a*e)^3*(d + e*x)^(3/2))/(3*e^7) + (6*b^4*(b*d - a*e)^2*(d + e*x)^(5/2))/e^
7 - (12*b^5*(b*d - a*e)*(d + e*x)^(7/2))/(7*e^7) + (2*b^6*(d + e*x)^(9/2))/(9*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*} \text {integral}& = \int \frac {(a+b x)^6}{(d+e x)^{5/2}} \, dx \\ & = \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{5/2}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{3/2}}+\frac {15 b^2 (b d-a e)^4}{e^6 \sqrt {d+e x}}-\frac {20 b^3 (b d-a e)^3 \sqrt {d+e x}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{5/2}}{e^6}+\frac {b^6 (d+e x)^{7/2}}{e^6}\right ) \, dx \\ & = -\frac {2 (b d-a e)^6}{3 e^7 (d+e x)^{3/2}}+\frac {12 b (b d-a e)^5}{e^7 \sqrt {d+e x}}+\frac {30 b^2 (b d-a e)^4 \sqrt {d+e x}}{e^7}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{3/2}}{3 e^7}+\frac {6 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^7}-\frac {12 b^5 (b d-a e) (d+e x)^{7/2}}{7 e^7}+\frac {2 b^6 (d+e x)^{9/2}}{9 e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \left (-21 a^6 e^6-126 a^5 b e^5 (2 d+3 e x)+315 a^4 b^2 e^4 \left (8 d^2+12 d e x+3 e^2 x^2\right )+420 a^3 b^3 e^3 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+63 a^2 b^4 e^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-18 a b^5 e \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )+b^6 \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )\right )}{63 e^7 (d+e x)^{3/2}} \]

[In]

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

[Out]

(2*(-21*a^6*e^6 - 126*a^5*b*e^5*(2*d + 3*e*x) + 315*a^4*b^2*e^4*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 420*a^3*b^3*e
^3*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + 63*a^2*b^4*e^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8
*d*e^3*x^3 + 3*e^4*x^4) - 18*a*b^5*e*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 -
3*e^5*x^5) + b^6*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 +
7*e^6*x^6)))/(63*e^7*(d + e*x)^(3/2))

Maple [A] (verified)

Time = 2.63 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.52

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (-\frac {1}{3} b^{6} x^{6}+a^{6}-\frac {18}{7} a \,x^{5} b^{5}-9 a^{2} x^{4} b^{4}-20 a^{3} x^{3} b^{3}-45 a^{4} x^{2} b^{2}+18 a^{5} x b \right ) e^{6}+12 b \left (\frac {1}{21} b^{5} x^{5}+\frac {3}{7} a \,b^{4} x^{4}+2 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}-15 a^{4} b x +a^{5}\right ) d \,e^{5}-120 b^{2} \left (\frac {1}{105} b^{4} x^{4}+\frac {4}{35} a \,b^{3} x^{3}+\frac {6}{5} a^{2} b^{2} x^{2}-4 a^{3} b x +a^{4}\right ) d^{2} e^{4}+320 b^{3} \left (\frac {1}{105} b^{3} x^{3}+\frac {9}{35} a \,b^{2} x^{2}-\frac {9}{5} a^{2} b x +a^{3}\right ) d^{3} e^{3}-384 \left (\frac {1}{21} b^{2} x^{2}-\frac {6}{7} a b x +a^{2}\right ) b^{4} d^{4} e^{2}+\frac {1536 \left (-\frac {b x}{3}+a \right ) b^{5} d^{5} e}{7}-\frac {1024 b^{6} d^{6}}{21}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\)	\(276\)
risch	\(\frac {2 b^{2} \left (7 b^{4} x^{4} e^{4}+54 x^{3} a \,b^{3} e^{4}-26 x^{3} b^{4} d \,e^{3}+189 x^{2} a^{2} b^{2} e^{4}-216 x^{2} a \,b^{3} d \,e^{3}+69 x^{2} b^{4} d^{2} e^{2}+420 x \,a^{3} b \,e^{4}-882 x \,a^{2} b^{2} d \,e^{3}+666 x a \,b^{3} d^{2} e^{2}-176 x \,b^{4} d^{3} e +945 e^{4} a^{4}-3360 b \,e^{3} d \,a^{3}+4599 b^{2} e^{2} d^{2} a^{2}-2844 a \,b^{3} d^{3} e +667 b^{4} d^{4}\right ) \sqrt {e x +d}}{63 e^{7}}-\frac {2 \left (18 b e x +a e +17 b d \right ) \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{3 e^{7} \left (e x +d \right )^{\frac {3}{2}}}\)	\(279\)
gosper	\(-\frac {2 \left (-7 x^{6} b^{6} e^{6}-54 x^{5} a \,b^{5} e^{6}+12 x^{5} b^{6} d \,e^{5}-189 x^{4} a^{2} b^{4} e^{6}+108 x^{4} a \,b^{5} d \,e^{5}-24 x^{4} b^{6} d^{2} e^{4}-420 x^{3} a^{3} b^{3} e^{6}+504 x^{3} a^{2} b^{4} d \,e^{5}-288 x^{3} a \,b^{5} d^{2} e^{4}+64 x^{3} b^{6} d^{3} e^{3}-945 x^{2} a^{4} b^{2} e^{6}+2520 x^{2} a^{3} b^{3} d \,e^{5}-3024 x^{2} a^{2} b^{4} d^{2} e^{4}+1728 x^{2} a \,b^{5} d^{3} e^{3}-384 x^{2} b^{6} d^{4} e^{2}+378 x \,a^{5} b \,e^{6}-3780 x \,a^{4} b^{2} d \,e^{5}+10080 x \,a^{3} b^{3} d^{2} e^{4}-12096 x \,a^{2} b^{4} d^{3} e^{3}+6912 x a \,b^{5} d^{4} e^{2}-1536 x \,b^{6} d^{5} e +21 a^{6} e^{6}+252 a^{5} b d \,e^{5}-2520 a^{4} b^{2} d^{2} e^{4}+6720 a^{3} b^{3} d^{3} e^{3}-8064 a^{2} b^{4} d^{4} e^{2}+4608 a \,b^{5} d^{5} e -1024 b^{6} d^{6}\right )}{63 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\)	\(377\)
trager	\(-\frac {2 \left (-7 x^{6} b^{6} e^{6}-54 x^{5} a \,b^{5} e^{6}+12 x^{5} b^{6} d \,e^{5}-189 x^{4} a^{2} b^{4} e^{6}+108 x^{4} a \,b^{5} d \,e^{5}-24 x^{4} b^{6} d^{2} e^{4}-420 x^{3} a^{3} b^{3} e^{6}+504 x^{3} a^{2} b^{4} d \,e^{5}-288 x^{3} a \,b^{5} d^{2} e^{4}+64 x^{3} b^{6} d^{3} e^{3}-945 x^{2} a^{4} b^{2} e^{6}+2520 x^{2} a^{3} b^{3} d \,e^{5}-3024 x^{2} a^{2} b^{4} d^{2} e^{4}+1728 x^{2} a \,b^{5} d^{3} e^{3}-384 x^{2} b^{6} d^{4} e^{2}+378 x \,a^{5} b \,e^{6}-3780 x \,a^{4} b^{2} d \,e^{5}+10080 x \,a^{3} b^{3} d^{2} e^{4}-12096 x \,a^{2} b^{4} d^{3} e^{3}+6912 x a \,b^{5} d^{4} e^{2}-1536 x \,b^{6} d^{5} e +21 a^{6} e^{6}+252 a^{5} b d \,e^{5}-2520 a^{4} b^{2} d^{2} e^{4}+6720 a^{3} b^{3} d^{3} e^{3}-8064 a^{2} b^{4} d^{4} e^{2}+4608 a \,b^{5} d^{5} e -1024 b^{6} d^{6}\right )}{63 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\)	\(377\)
derivativedivides	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {12 a \,b^{5} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 b^{6} d \left (e x +d \right )^{\frac {7}{2}}}{7}+6 a^{2} b^{4} e^{2} \left (e x +d \right )^{\frac {5}{2}}-12 a \,b^{5} d e \left (e x +d \right )^{\frac {5}{2}}+6 b^{6} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {40 a^{3} b^{3} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-40 a^{2} b^{4} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+40 a \,b^{5} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {40 b^{6} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+30 a^{4} b^{2} e^{4} \sqrt {e x +d}-120 a^{3} b^{3} d \,e^{3} \sqrt {e x +d}+180 a^{2} b^{4} d^{2} e^{2} \sqrt {e x +d}-120 a \,b^{5} d^{3} e \sqrt {e x +d}+30 b^{6} d^{4} \sqrt {e x +d}-\frac {12 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(413\)
default	\(\frac {\frac {2 b^{6} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {12 a \,b^{5} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 b^{6} d \left (e x +d \right )^{\frac {7}{2}}}{7}+6 a^{2} b^{4} e^{2} \left (e x +d \right )^{\frac {5}{2}}-12 a \,b^{5} d e \left (e x +d \right )^{\frac {5}{2}}+6 b^{6} d^{2} \left (e x +d \right )^{\frac {5}{2}}+\frac {40 a^{3} b^{3} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-40 a^{2} b^{4} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+40 a \,b^{5} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {40 b^{6} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+30 a^{4} b^{2} e^{4} \sqrt {e x +d}-120 a^{3} b^{3} d \,e^{3} \sqrt {e x +d}+180 a^{2} b^{4} d^{2} e^{2} \sqrt {e x +d}-120 a \,b^{5} d^{3} e \sqrt {e x +d}+30 b^{6} d^{4} \sqrt {e x +d}-\frac {12 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\)	\(413\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*((-1/3*b^6*x^6+a^6-18/7*a*x^5*b^5-9*a^2*x^4*b^4-20*a^3*x^3*b^3-45*a^4*x^2*b^2+18*a^5*x*b)*e^6+12*b*(1/21*
b^5*x^5+3/7*a*b^4*x^4+2*a^2*b^3*x^3+10*a^3*b^2*x^2-15*a^4*b*x+a^5)*d*e^5-120*b^2*(1/105*b^4*x^4+4/35*a*b^3*x^3
+6/5*a^2*b^2*x^2-4*a^3*b*x+a^4)*d^2*e^4+320*b^3*(1/105*b^3*x^3+9/35*a*b^2*x^2-9/5*a^2*b*x+a^3)*d^3*e^3-384*(1/
21*b^2*x^2-6/7*a*b*x+a^2)*b^4*d^4*e^2+1536/7*(-1/3*b*x+a)*b^5*d^5*e-1024/21*b^6*d^6)/(e*x+d)^(3/2)/e^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (159) = 318\).

Time = 0.27 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.08 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 4608 \, a b^{5} d^{5} e + 8064 \, a^{2} b^{4} d^{4} e^{2} - 6720 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} - 252 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} - 6 \, {\left (2 \, b^{6} d e^{5} - 9 \, a b^{5} e^{6}\right )} x^{5} + 3 \, {\left (8 \, b^{6} d^{2} e^{4} - 36 \, a b^{5} d e^{5} + 63 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \, {\left (16 \, b^{6} d^{3} e^{3} - 72 \, a b^{5} d^{2} e^{4} + 126 \, a^{2} b^{4} d e^{5} - 105 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{4} e^{2} - 576 \, a b^{5} d^{3} e^{3} + 1008 \, a^{2} b^{4} d^{2} e^{4} - 840 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (256 \, b^{6} d^{5} e - 1152 \, a b^{5} d^{4} e^{2} + 2016 \, a^{2} b^{4} d^{3} e^{3} - 1680 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} - 63 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*b^6*e^6*x^6 + 1024*b^6*d^6 - 4608*a*b^5*d^5*e + 8064*a^2*b^4*d^4*e^2 - 6720*a^3*b^3*d^3*e^3 + 2520*a^4
*b^2*d^2*e^4 - 252*a^5*b*d*e^5 - 21*a^6*e^6 - 6*(2*b^6*d*e^5 - 9*a*b^5*e^6)*x^5 + 3*(8*b^6*d^2*e^4 - 36*a*b^5*
d*e^5 + 63*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 72*a*b^5*d^2*e^4 + 126*a^2*b^4*d*e^5 - 105*a^3*b^3*e^6)*x^3
+ 3*(128*b^6*d^4*e^2 - 576*a*b^5*d^3*e^3 + 1008*a^2*b^4*d^2*e^4 - 840*a^3*b^3*d*e^5 + 315*a^4*b^2*e^6)*x^2 + 6
*(256*b^6*d^5*e - 1152*a*b^5*d^4*e^2 + 2016*a^2*b^4*d^3*e^3 - 1680*a^3*b^3*d^2*e^4 + 630*a^4*b^2*d*e^5 - 63*a^
5*b*e^6)*x)*sqrt(e*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (168) = 336\).

Time = 10.37 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{6} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} - \frac {6 b \left (a e - b d\right )^{5}}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a b^{5} e - 6 b^{6} d\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (15 a^{2} b^{4} e^{2} - 30 a b^{5} d e + 15 b^{6} d^{2}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (20 a^{3} b^{3} e^{3} - 60 a^{2} b^{4} d e^{2} + 60 a b^{5} d^{2} e - 20 b^{6} d^{3}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (15 a^{4} b^{2} e^{4} - 60 a^{3} b^{3} d e^{3} + 90 a^{2} b^{4} d^{2} e^{2} - 60 a b^{5} d^{3} e + 15 b^{6} d^{4}\right )}{e^{6}} - \frac {\left (a e - b d\right )^{6}}{3 e^{6} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(5/2),x)

[Out]

Piecewise((2*(b**6*(d + e*x)**(9/2)/(9*e**6) - 6*b*(a*e - b*d)**5/(e**6*sqrt(d + e*x)) + (d + e*x)**(7/2)*(6*a
*b**5*e - 6*b**6*d)/(7*e**6) + (d + e*x)**(5/2)*(15*a**2*b**4*e**2 - 30*a*b**5*d*e + 15*b**6*d**2)/(5*e**6) +
(d + e*x)**(3/2)*(20*a**3*b**3*e**3 - 60*a**2*b**4*d*e**2 + 60*a*b**5*d**2*e - 20*b**6*d**3)/(3*e**6) + sqrt(d
+ e*x)*(15*a**4*b**2*e**4 - 60*a**3*b**3*d*e**3 + 90*a**2*b**4*d**2*e**2 - 60*a*b**5*d**3*e + 15*b**6*d**4)/e
**6 - (a*e - b*d)**6/(3*e**6*(d + e*x)**(3/2)))/e, Ne(e, 0)), ((a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*x**3 + 5*
a**3*b**3*x**4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7)/d**(5/2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (159) = 318\).

Time = 0.22 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {7 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{6} - 54 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 420 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 945 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {21 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6} - 18 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{6}}\right )}}{63 \, e} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/63*((7*(e*x + d)^(9/2)*b^6 - 54*(b^6*d - a*b^5*e)*(e*x + d)^(7/2) + 189*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2
)*(e*x + d)^(5/2) - 420*(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)*(e*x + d)^(3/2) + 945*(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)*sqrt(e*x + d))/e^6 - 21*(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 - 18*(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)*(e*x + d))/((e*x
+ d)^(3/2)*e^6))/e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (159) = 318\).

Time = 0.29 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (18 \, {\left (e x + d\right )} b^{6} d^{5} - b^{6} d^{6} - 90 \, {\left (e x + d\right )} a b^{5} d^{4} e + 6 \, a b^{5} d^{5} e + 180 \, {\left (e x + d\right )} a^{2} b^{4} d^{3} e^{2} - 15 \, a^{2} b^{4} d^{4} e^{2} - 180 \, {\left (e x + d\right )} a^{3} b^{3} d^{2} e^{3} + 20 \, a^{3} b^{3} d^{3} e^{3} + 90 \, {\left (e x + d\right )} a^{4} b^{2} d e^{4} - 15 \, a^{4} b^{2} d^{2} e^{4} - 18 \, {\left (e x + d\right )} a^{5} b e^{5} + 6 \, a^{5} b d e^{5} - a^{6} e^{6}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{7}} + \frac {2 \, {\left (7 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{6} e^{56} - 54 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} d e^{56} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d^{2} e^{56} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{3} e^{56} + 945 \, \sqrt {e x + d} b^{6} d^{4} e^{56} + 54 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{5} e^{57} - 378 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} d e^{57} + 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d^{2} e^{57} - 3780 \, \sqrt {e x + d} a b^{5} d^{3} e^{57} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{4} e^{58} - 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} d e^{58} + 5670 \, \sqrt {e x + d} a^{2} b^{4} d^{2} e^{58} + 420 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{3} e^{59} - 3780 \, \sqrt {e x + d} a^{3} b^{3} d e^{59} + 945 \, \sqrt {e x + d} a^{4} b^{2} e^{60}\right )}}{63 \, e^{63}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/3*(18*(e*x + d)*b^6*d^5 - b^6*d^6 - 90*(e*x + d)*a*b^5*d^4*e + 6*a*b^5*d^5*e + 180*(e*x + d)*a^2*b^4*d^3*e^2
- 15*a^2*b^4*d^4*e^2 - 180*(e*x + d)*a^3*b^3*d^2*e^3 + 20*a^3*b^3*d^3*e^3 + 90*(e*x + d)*a^4*b^2*d*e^4 - 15*a
^4*b^2*d^2*e^4 - 18*(e*x + d)*a^5*b*e^5 + 6*a^5*b*d*e^5 - a^6*e^6)/((e*x + d)^(3/2)*e^7) + 2/63*(7*(e*x + d)^(
9/2)*b^6*e^56 - 54*(e*x + d)^(7/2)*b^6*d*e^56 + 189*(e*x + d)^(5/2)*b^6*d^2*e^56 - 420*(e*x + d)^(3/2)*b^6*d^3
*e^56 + 945*sqrt(e*x + d)*b^6*d^4*e^56 + 54*(e*x + d)^(7/2)*a*b^5*e^57 - 378*(e*x + d)^(5/2)*a*b^5*d*e^57 + 12
60*(e*x + d)^(3/2)*a*b^5*d^2*e^57 - 3780*sqrt(e*x + d)*a*b^5*d^3*e^57 + 189*(e*x + d)^(5/2)*a^2*b^4*e^58 - 126
0*(e*x + d)^(3/2)*a^2*b^4*d*e^58 + 5670*sqrt(e*x + d)*a^2*b^4*d^2*e^58 + 420*(e*x + d)^(3/2)*a^3*b^3*e^59 - 37
80*sqrt(e*x + d)*a^3*b^3*d*e^59 + 945*sqrt(e*x + d)*a^4*b^2*e^60)/e^63

Mupad [B] (verification not implemented)

Time = 9.79 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2\,b^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {\left (d+e\,x\right )\,\left (-12\,a^5\,b\,e^5+60\,a^4\,b^2\,d\,e^4-120\,a^3\,b^3\,d^2\,e^3+120\,a^2\,b^4\,d^3\,e^2-60\,a\,b^5\,d^4\,e+12\,b^6\,d^5\right )-\frac {2\,a^6\,e^6}{3}-\frac {2\,b^6\,d^6}{3}-10\,a^2\,b^4\,d^4\,e^2+\frac {40\,a^3\,b^3\,d^3\,e^3}{3}-10\,a^4\,b^2\,d^2\,e^4+4\,a\,b^5\,d^5\,e+4\,a^5\,b\,d\,e^5}{e^7\,{\left (d+e\,x\right )}^{3/2}}+\frac {30\,b^2\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^7}+\frac {40\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {6\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{e^7} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^(5/2),x)

[Out]

(2*b^6*(d + e*x)^(9/2))/(9*e^7) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(7/2))/(7*e^7) + ((d + e*x)*(12*b^6*d^5 -
12*a^5*b*e^5 + 60*a^4*b^2*d*e^4 + 120*a^2*b^4*d^3*e^2 - 120*a^3*b^3*d^2*e^3 - 60*a*b^5*d^4*e) - (2*a^6*e^6)/3
- (2*b^6*d^6)/3 - 10*a^2*b^4*d^4*e^2 + (40*a^3*b^3*d^3*e^3)/3 - 10*a^4*b^2*d^2*e^4 + 4*a*b^5*d^5*e + 4*a^5*b*
d*e^5)/(e^7*(d + e*x)^(3/2)) + (30*b^2*(a*e - b*d)^4*(d + e*x)^(1/2))/e^7 + (40*b^3*(a*e - b*d)^3*(d + e*x)^(3
/2))/(3*e^7) + (6*b^4*(a*e - b*d)^2*(d + e*x)^(5/2))/e^7