\(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1693]

Optimal result

Integrand size = 30, antiderivative size = 320 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^6 (a+b x)} \]

[Out]

-2/5*(-a*e+b*d)^5*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+10/7*b*(-a*e+b*d)^4*(e*x+d)^(7/2)*((b*x+a)^2)^(1
/2)/e^6/(b*x+a)-20/9*b^2*(-a*e+b*d)^3*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+20/11*b^3*(-a*e+b*d)^2*(e*x+
d)^(11/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-10/13*b^4*(-a*e+b*d)*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+2/15
*b^5*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45} \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^6 (a+b x)} \]

[In]

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^6*(a + b*x)) + (10*b*(b*d - a*e)^4*(d +
e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(9/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(11*e^6*(a + b*x)) - (10*b^4*(b*d - a*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^6*(a + b*x)) +
(2*b^5*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^6*(a + b*x))

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])

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (d+e x)^{3/2} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (d+e x)^{3/2}}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{5/2}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{7/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{9/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{11/2}}{e^5}+\frac {b^{10} (d+e x)^{13/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {2 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^6 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.73 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (9009 a^5 e^5+6435 a^4 b e^4 (-2 d+5 e x)+1430 a^3 b^2 e^3 \left (8 d^2-20 d e x+35 e^2 x^2\right )+390 a^2 b^3 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+15 a b^4 e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^5 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )}{45045 e^6 (a+b x)} \]

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(9009*a^5*e^5 + 6435*a^4*b*e^4*(-2*d + 5*e*x) + 1430*a^3*b^2*e^3*(8*d^2 -
20*d*e*x + 35*e^2*x^2) + 390*a^2*b^3*e^2*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 15*a*b^4*e*(12
8*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) + b^5*(-256*d^5 + 640*d^4*e*x - 1120*d^3
*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^5)))/(45045*e^6*(a + b*x))

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.90

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 x^{5} e^{5} b^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 b^{5} d^{4} e x +9009 a^{5} e^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{6} \left (b x +a \right )^{5}}\)	\(289\)
default	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 x^{5} e^{5} b^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 b^{5} d^{4} e x +9009 a^{5} e^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{6} \left (b x +a \right )^{5}}\)	\(289\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{5} x^{7} e^{7}+17325 a \,b^{4} e^{7} x^{6}+3696 b^{5} d \,e^{6} x^{6}+40950 a^{2} b^{3} e^{7} x^{5}+22050 a \,b^{4} d \,e^{6} x^{5}+63 b^{5} d^{2} e^{5} x^{5}+50050 a^{3} b^{2} e^{7} x^{4}+54600 a^{2} b^{3} d \,e^{6} x^{4}+525 a \,b^{4} d^{2} e^{5} x^{4}-70 b^{5} d^{3} e^{4} x^{4}+32175 a^{4} b \,e^{7} x^{3}+71500 a^{3} b^{2} d \,e^{6} x^{3}+1950 a^{2} b^{3} d^{2} e^{5} x^{3}-600 a \,b^{4} d^{3} e^{4} x^{3}+80 b^{5} d^{4} e^{3} x^{3}+9009 a^{5} e^{7} x^{2}+51480 a^{4} b d \,e^{6} x^{2}+4290 a^{3} b^{2} d^{2} e^{5} x^{2}-2340 a^{2} b^{3} d^{3} e^{4} x^{2}+720 a \,b^{4} d^{4} e^{3} x^{2}-96 b^{5} d^{5} e^{2} x^{2}+18018 a^{5} d \,e^{6} x +6435 a^{4} b \,d^{2} e^{5} x -5720 a^{3} b^{2} d^{3} e^{4} x +3120 a^{2} b^{3} d^{4} e^{3} x -960 a \,b^{4} d^{5} e^{2} x +128 b^{5} d^{6} e x +9009 a^{5} d^{2} e^{5}-12870 a^{4} b \,d^{3} e^{4}+11440 a^{3} b^{2} d^{4} e^{3}-6240 a^{2} b^{3} d^{5} e^{2}+1920 a \,b^{4} d^{6} e -256 b^{5} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{6}}\)	\(469\)

[In]

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

[Out]

2/45045*(e*x+d)^(5/2)*(3003*b^5*e^5*x^5+17325*a*b^4*e^5*x^4-2310*b^5*d*e^4*x^4+40950*a^2*b^3*e^5*x^3-12600*a*b
^4*d*e^4*x^3+1680*b^5*d^2*e^3*x^3+50050*a^3*b^2*e^5*x^2-27300*a^2*b^3*d*e^4*x^2+8400*a*b^4*d^2*e^3*x^2-1120*b^
5*d^3*e^2*x^2+32175*a^4*b*e^5*x-28600*a^3*b^2*d*e^4*x+15600*a^2*b^3*d^2*e^3*x-4800*a*b^4*d^3*e^2*x+640*b^5*d^4
*e*x+9009*a^5*e^5-12870*a^4*b*d*e^4+11440*a^3*b^2*d^2*e^3-6240*a^2*b^3*d^3*e^2+1920*a*b^4*d^4*e-256*b^5*d^5)*(
(b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.31 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]

[In]

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

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12
870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e
^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4
+ 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*
(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^
5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2
*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)/e^6

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.31 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]

[In]

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

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12
870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e
^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4
+ 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*
(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^
5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2
*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)/e^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (230) = 460\).

Time = 0.36 (sec) , antiderivative size = 1192, normalized size of antiderivative = 3.72 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/45045*(45045*sqrt(e*x + d)*a^5*d^2*sgn(b*x + a) + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^5*d*sgn(b*x
+ a) + 75075*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*b*d^2*sgn(b*x + a)/e + 3003*(3*(e*x + d)^(5/2) - 10*(e*
x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*sgn(b*x + a) + 30030*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15
*sqrt(e*x + d)*d^2)*a^3*b^2*d^2*sgn(b*x + a)/e^2 + 30030*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e
*x + d)*d^2)*a^4*b*d*sgn(b*x + a)/e + 12870*(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^3*d^2*sgn(b*x + a)/e^3 + 25740*(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^2*d*sgn(b*x + a)/e^2 + 6435*(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*b*sgn(b*x + a)/e + 715*(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^4*d^2*sgn(
b*x + a)/e^4 + 2860*(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^3*d*sgn(b*x + a)/e^3 + 1430*(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^2*sgn(b*x + a)/e^2 + 65*(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^5*d^2*sgn(b*x + a)/e^5 + 650*(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*b
^4*d*sgn(b*x + a)/e^4 + 650*(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^3*sgn(b*x + a)/e^3 + 30*(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^5*d*sgn(b*x + a)/e^5 + 75*(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 - 6
006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*a*b^4*sgn(b*x + a)/e^4 + 7*(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^5*sgn(b*x + a)/e^5)/e

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

[In]

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

[Out]

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