3.1991 \(\int (a+b x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=311 \[ \frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11} (b d-a e)}{2 b^7}+\frac{15 e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)^2}{11 b^7}+\frac{2 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^3}{b^7}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^4}{3 b^7}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^5}{4 b^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^6}{7 b^7}+\frac{e^6 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{12}}{13 b^7} \]

[Out]

((b*d - a*e)^6*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^7) + (3*e*(b*d - a*e)^5*(a + b*x)^7*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*b^7) + (5*e^2*(b*d - a*e)^4*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^7) + (2*e^3
*(b*d - a*e)^3*(a + b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^7 + (15*e^4*(b*d - a*e)^2*(a + b*x)^10*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(11*b^7) + (e^5*(b*d - a*e)*(a + b*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^7) + (e^6*(a
+ b*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*b^7)

________________________________________________________________________________________

Rubi [A]  time = 0.44672, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{e^5 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{11} (b d-a e)}{2 b^7}+\frac{15 e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)^2}{11 b^7}+\frac{2 e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^3}{b^7}+\frac{5 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^4}{3 b^7}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^5}{4 b^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^6}{7 b^7}+\frac{e^6 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{12}}{13 b^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^6*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^7) + (3*e*(b*d - a*e)^5*(a + b*x)^7*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*b^7) + (5*e^2*(b*d - a*e)^4*(a + b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^7) + (2*e^3
*(b*d - a*e)^3*(a + b*x)^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^7 + (15*e^4*(b*d - a*e)^2*(a + b*x)^10*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(11*b^7) + (e^5*(b*d - a*e)*(a + b*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^7) + (e^6*(a
+ b*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*b^7)

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

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 (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^6 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^6 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(b d-a e)^6 (a+b x)^6}{b^6}+\frac{6 e (b d-a e)^5 (a+b x)^7}{b^6}+\frac{15 e^2 (b d-a e)^4 (a+b x)^8}{b^6}+\frac{20 e^3 (b d-a e)^3 (a+b x)^9}{b^6}+\frac{15 e^4 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac{6 e^5 (b d-a e) (a+b x)^{11}}{b^6}+\frac{e^6 (a+b x)^{12}}{b^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^6 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^7}+\frac{3 e (b d-a e)^5 (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^7}+\frac{5 e^2 (b d-a e)^4 (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{3 b^7}+\frac{2 e^3 (b d-a e)^3 (a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{b^7}+\frac{15 e^4 (b d-a e)^2 (a+b x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{11 b^7}+\frac{e^5 (b d-a e) (a+b x)^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^7}+\frac{e^6 (a+b x)^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{13 b^7}\\ \end{align*}

Mathematica [A]  time = 0.165768, size = 525, normalized size = 1.69 \[ \frac{x \sqrt{(a+b x)^2} \left (715 a^4 b^2 x^2 \left (756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+378 d^5 e x+84 d^6+189 d e^5 x^5+28 e^6 x^6\right )+286 a^3 b^3 x^3 \left (2100 d^4 e^2 x^2+2400 d^3 e^3 x^3+1575 d^2 e^4 x^4+1008 d^5 e x+210 d^6+560 d e^5 x^5+84 e^6 x^6\right )+78 a^2 b^4 x^4 \left (4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+2310 d^5 e x+462 d^6+1386 d e^5 x^5+210 e^6 x^6\right )+1287 a^5 b x \left (210 d^4 e^2 x^2+224 d^3 e^3 x^3+140 d^2 e^4 x^4+112 d^5 e x+28 d^6+48 d e^5 x^5+7 e^6 x^6\right )+1716 a^6 \left (35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+21 d^5 e x+7 d^6+7 d e^5 x^5+e^6 x^6\right )+13 a b^5 x^5 \left (10395 d^4 e^2 x^2+12320 d^3 e^3 x^3+8316 d^2 e^4 x^4+4752 d^5 e x+924 d^6+3024 d e^5 x^5+462 e^6 x^6\right )+b^6 x^6 \left (20020 d^4 e^2 x^2+24024 d^3 e^3 x^3+16380 d^2 e^4 x^4+9009 d^5 e x+1716 d^6+6006 d e^5 x^5+924 e^6 x^6\right )\right )}{12012 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(1716*a^6*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 21*d^2*e^4*x^4 + 7*d*e^
5*x^5 + e^6*x^6) + 1287*a^5*b*x*(28*d^6 + 112*d^5*e*x + 210*d^4*e^2*x^2 + 224*d^3*e^3*x^3 + 140*d^2*e^4*x^4 +
48*d*e^5*x^5 + 7*e^6*x^6) + 715*a^4*b^2*x^2*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^
2*e^4*x^4 + 189*d*e^5*x^5 + 28*e^6*x^6) + 286*a^3*b^3*x^3*(210*d^6 + 1008*d^5*e*x + 2100*d^4*e^2*x^2 + 2400*d^
3*e^3*x^3 + 1575*d^2*e^4*x^4 + 560*d*e^5*x^5 + 84*e^6*x^6) + 78*a^2*b^4*x^4*(462*d^6 + 2310*d^5*e*x + 4950*d^4
*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 + 1386*d*e^5*x^5 + 210*e^6*x^6) + 13*a*b^5*x^5*(924*d^6 + 4752*
d^5*e*x + 10395*d^4*e^2*x^2 + 12320*d^3*e^3*x^3 + 8316*d^2*e^4*x^4 + 3024*d*e^5*x^5 + 462*e^6*x^6) + b^6*x^6*(
1716*d^6 + 9009*d^5*e*x + 20020*d^4*e^2*x^2 + 24024*d^3*e^3*x^3 + 16380*d^2*e^4*x^4 + 6006*d*e^5*x^5 + 924*e^6
*x^6)))/(12012*(a + b*x))

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 707, normalized size = 2.3 \begin{align*}{\frac{x \left ( 924\,{b}^{6}{e}^{6}{x}^{12}+6006\,{x}^{11}{b}^{5}a{e}^{6}+6006\,{x}^{11}{b}^{6}d{e}^{5}+16380\,{x}^{10}{a}^{2}{b}^{4}{e}^{6}+39312\,{x}^{10}{b}^{5}ad{e}^{5}+16380\,{x}^{10}{b}^{6}{d}^{2}{e}^{4}+24024\,{a}^{3}{b}^{3}{e}^{6}{x}^{9}+108108\,{a}^{2}{b}^{4}d{e}^{5}{x}^{9}+108108\,a{b}^{5}{d}^{2}{e}^{4}{x}^{9}+24024\,{b}^{6}{d}^{3}{e}^{3}{x}^{9}+20020\,{x}^{8}{a}^{4}{b}^{2}{e}^{6}+160160\,{x}^{8}{a}^{3}{b}^{3}d{e}^{5}+300300\,{x}^{8}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+160160\,{x}^{8}{b}^{5}a{d}^{3}{e}^{3}+20020\,{x}^{8}{b}^{6}{d}^{4}{e}^{2}+9009\,{x}^{7}{a}^{5}b{e}^{6}+135135\,{x}^{7}{a}^{4}{b}^{2}d{e}^{5}+450450\,{x}^{7}{a}^{3}{b}^{3}{d}^{2}{e}^{4}+450450\,{x}^{7}{a}^{2}{b}^{4}{d}^{3}{e}^{3}+135135\,{x}^{7}{b}^{5}a{d}^{4}{e}^{2}+9009\,{x}^{7}{b}^{6}{d}^{5}e+1716\,{x}^{6}{a}^{6}{e}^{6}+61776\,{x}^{6}{a}^{5}bd{e}^{5}+386100\,{x}^{6}{a}^{4}{b}^{2}{d}^{2}{e}^{4}+686400\,{x}^{6}{a}^{3}{b}^{3}{d}^{3}{e}^{3}+386100\,{x}^{6}{a}^{2}{b}^{4}{d}^{4}{e}^{2}+61776\,{x}^{6}{b}^{5}a{d}^{5}e+1716\,{x}^{6}{b}^{6}{d}^{6}+12012\,{a}^{6}d{e}^{5}{x}^{5}+180180\,{a}^{5}b{d}^{2}{e}^{4}{x}^{5}+600600\,{a}^{4}{b}^{2}{d}^{3}{e}^{3}{x}^{5}+600600\,{a}^{3}{b}^{3}{d}^{4}{e}^{2}{x}^{5}+180180\,{a}^{2}{b}^{4}{d}^{5}e{x}^{5}+12012\,a{b}^{5}{d}^{6}{x}^{5}+36036\,{a}^{6}{d}^{2}{e}^{4}{x}^{4}+288288\,{a}^{5}b{d}^{3}{e}^{3}{x}^{4}+540540\,{a}^{4}{b}^{2}{d}^{4}{e}^{2}{x}^{4}+288288\,{a}^{3}{b}^{3}{d}^{5}e{x}^{4}+36036\,{a}^{2}{b}^{4}{d}^{6}{x}^{4}+60060\,{x}^{3}{a}^{6}{d}^{3}{e}^{3}+270270\,{x}^{3}{a}^{5}b{d}^{4}{e}^{2}+270270\,{x}^{3}{a}^{4}{b}^{2}{d}^{5}e+60060\,{x}^{3}{a}^{3}{b}^{3}{d}^{6}+60060\,{a}^{6}{d}^{4}{e}^{2}{x}^{2}+144144\,{a}^{5}b{d}^{5}e{x}^{2}+60060\,{a}^{4}{b}^{2}{d}^{6}{x}^{2}+36036\,{a}^{6}{d}^{5}ex+36036\,{a}^{5}b{d}^{6}x+12012\,{a}^{6}{d}^{6} \right ) }{12012\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/12012*x*(924*b^6*e^6*x^12+6006*a*b^5*e^6*x^11+6006*b^6*d*e^5*x^11+16380*a^2*b^4*e^6*x^10+39312*a*b^5*d*e^5*x
^10+16380*b^6*d^2*e^4*x^10+24024*a^3*b^3*e^6*x^9+108108*a^2*b^4*d*e^5*x^9+108108*a*b^5*d^2*e^4*x^9+24024*b^6*d
^3*e^3*x^9+20020*a^4*b^2*e^6*x^8+160160*a^3*b^3*d*e^5*x^8+300300*a^2*b^4*d^2*e^4*x^8+160160*a*b^5*d^3*e^3*x^8+
20020*b^6*d^4*e^2*x^8+9009*a^5*b*e^6*x^7+135135*a^4*b^2*d*e^5*x^7+450450*a^3*b^3*d^2*e^4*x^7+450450*a^2*b^4*d^
3*e^3*x^7+135135*a*b^5*d^4*e^2*x^7+9009*b^6*d^5*e*x^7+1716*a^6*e^6*x^6+61776*a^5*b*d*e^5*x^6+386100*a^4*b^2*d^
2*e^4*x^6+686400*a^3*b^3*d^3*e^3*x^6+386100*a^2*b^4*d^4*e^2*x^6+61776*a*b^5*d^5*e*x^6+1716*b^6*d^6*x^6+12012*a
^6*d*e^5*x^5+180180*a^5*b*d^2*e^4*x^5+600600*a^4*b^2*d^3*e^3*x^5+600600*a^3*b^3*d^4*e^2*x^5+180180*a^2*b^4*d^5
*e*x^5+12012*a*b^5*d^6*x^5+36036*a^6*d^2*e^4*x^4+288288*a^5*b*d^3*e^3*x^4+540540*a^4*b^2*d^4*e^2*x^4+288288*a^
3*b^3*d^5*e*x^4+36036*a^2*b^4*d^6*x^4+60060*a^6*d^3*e^3*x^3+270270*a^5*b*d^4*e^2*x^3+270270*a^4*b^2*d^5*e*x^3+
60060*a^3*b^3*d^6*x^3+60060*a^6*d^4*e^2*x^2+144144*a^5*b*d^5*e*x^2+60060*a^4*b^2*d^6*x^2+36036*a^6*d^5*e*x+360
36*a^5*b*d^6*x+12012*a^6*d^6)*((b*x+a)^2)^(5/2)/(b*x+a)^5

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.53286, size = 1227, normalized size = 3.95 \begin{align*} \frac{1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac{1}{2} \,{\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac{3}{11} \,{\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} +{\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac{5}{3} \,{\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac{3}{4} \,{\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} +{\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \,{\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac{5}{2} \,{\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \,{\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/13*b^6*e^6*x^13 + a^6*d^6*x + 1/2*(b^6*d*e^5 + a*b^5*e^6)*x^12 + 3/11*(5*b^6*d^2*e^4 + 12*a*b^5*d*e^5 + 5*a^
2*b^4*e^6)*x^11 + (2*b^6*d^3*e^3 + 9*a*b^5*d^2*e^4 + 9*a^2*b^4*d*e^5 + 2*a^3*b^3*e^6)*x^10 + 5/3*(b^6*d^4*e^2
+ 8*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^9 + 3/4*(b^6*d^5*e + 15*a*b^5*d^4*e^
2 + 50*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + a^5*b*e^6)*x^8 + 1/7*(b^6*d^6 + 36*a*b^5*d^5*
e + 225*a^2*b^4*d^4*e^2 + 400*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 36*a^5*b*d*e^5 + a^6*e^6)*x^7 + (a*b^5*d
^6 + 15*a^2*b^4*d^5*e + 50*a^3*b^3*d^4*e^2 + 50*a^4*b^2*d^3*e^3 + 15*a^5*b*d^2*e^4 + a^6*d*e^5)*x^6 + 3*(a^2*b
^4*d^6 + 8*a^3*b^3*d^5*e + 15*a^4*b^2*d^4*e^2 + 8*a^5*b*d^3*e^3 + a^6*d^2*e^4)*x^5 + 5/2*(2*a^3*b^3*d^6 + 9*a^
4*b^2*d^5*e + 9*a^5*b*d^4*e^2 + 2*a^6*d^3*e^3)*x^4 + (5*a^4*b^2*d^6 + 12*a^5*b*d^5*e + 5*a^6*d^4*e^2)*x^3 + 3*
(a^5*b*d^6 + a^6*d^5*e)*x^2

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**6*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.15376, size = 1289, normalized size = 4.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^6*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/13*b^6*x^13*e^6*sgn(b*x + a) + 1/2*b^6*d*x^12*e^5*sgn(b*x + a) + 15/11*b^6*d^2*x^11*e^4*sgn(b*x + a) + 2*b^6
*d^3*x^10*e^3*sgn(b*x + a) + 5/3*b^6*d^4*x^9*e^2*sgn(b*x + a) + 3/4*b^6*d^5*x^8*e*sgn(b*x + a) + 1/7*b^6*d^6*x
^7*sgn(b*x + a) + 1/2*a*b^5*x^12*e^6*sgn(b*x + a) + 36/11*a*b^5*d*x^11*e^5*sgn(b*x + a) + 9*a*b^5*d^2*x^10*e^4
*sgn(b*x + a) + 40/3*a*b^5*d^3*x^9*e^3*sgn(b*x + a) + 45/4*a*b^5*d^4*x^8*e^2*sgn(b*x + a) + 36/7*a*b^5*d^5*x^7
*e*sgn(b*x + a) + a*b^5*d^6*x^6*sgn(b*x + a) + 15/11*a^2*b^4*x^11*e^6*sgn(b*x + a) + 9*a^2*b^4*d*x^10*e^5*sgn(
b*x + a) + 25*a^2*b^4*d^2*x^9*e^4*sgn(b*x + a) + 75/2*a^2*b^4*d^3*x^8*e^3*sgn(b*x + a) + 225/7*a^2*b^4*d^4*x^7
*e^2*sgn(b*x + a) + 15*a^2*b^4*d^5*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^6*x^5*sgn(b*x + a) + 2*a^3*b^3*x^10*e^6*sg
n(b*x + a) + 40/3*a^3*b^3*d*x^9*e^5*sgn(b*x + a) + 75/2*a^3*b^3*d^2*x^8*e^4*sgn(b*x + a) + 400/7*a^3*b^3*d^3*x
^7*e^3*sgn(b*x + a) + 50*a^3*b^3*d^4*x^6*e^2*sgn(b*x + a) + 24*a^3*b^3*d^5*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^6*
x^4*sgn(b*x + a) + 5/3*a^4*b^2*x^9*e^6*sgn(b*x + a) + 45/4*a^4*b^2*d*x^8*e^5*sgn(b*x + a) + 225/7*a^4*b^2*d^2*
x^7*e^4*sgn(b*x + a) + 50*a^4*b^2*d^3*x^6*e^3*sgn(b*x + a) + 45*a^4*b^2*d^4*x^5*e^2*sgn(b*x + a) + 45/2*a^4*b^
2*d^5*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^6*x^3*sgn(b*x + a) + 3/4*a^5*b*x^8*e^6*sgn(b*x + a) + 36/7*a^5*b*d*x^7*
e^5*sgn(b*x + a) + 15*a^5*b*d^2*x^6*e^4*sgn(b*x + a) + 24*a^5*b*d^3*x^5*e^3*sgn(b*x + a) + 45/2*a^5*b*d^4*x^4*
e^2*sgn(b*x + a) + 12*a^5*b*d^5*x^3*e*sgn(b*x + a) + 3*a^5*b*d^6*x^2*sgn(b*x + a) + 1/7*a^6*x^7*e^6*sgn(b*x +
a) + a^6*d*x^6*e^5*sgn(b*x + a) + 3*a^6*d^2*x^5*e^4*sgn(b*x + a) + 5*a^6*d^3*x^4*e^3*sgn(b*x + a) + 5*a^6*d^4*
x^3*e^2*sgn(b*x + a) + 3*a^6*d^5*x^2*e*sgn(b*x + a) + a^6*d^6*x*sgn(b*x + a)