3.18.66 \(\int (a+b x) (d+e x)^6 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)
Optimal. Leaf size=311 \[ \frac {e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11} (b d-a e)}{2 b^7}+\frac {15 e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)^2}{11 b^7}+\frac {2 e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^3}{b^7}+\frac {5 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^4}{3 b^7}+\frac {3 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^5}{4 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^6}{7 b^7}+\frac {e^6 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{12}}{13 b^7} \]
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Rubi [A] time = 0.45, antiderivative size = 311, normalized size of antiderivative = 1.00,
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} \begin {gather*} \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} \end {gather*}
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 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])
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]
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*}
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Mathematica [A] time = 0.16, size = 525, normalized size = 1.69 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (1716 a^6 \left (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\right )+1287 a^5 b x \left (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\right )+715 a^4 b^2 x^2 \left (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\right )+286 a^3 b^3 x^3 \left (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\right )+78 a^2 b^4 x^4 \left (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\right )+13 a b^5 x^5 \left (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\right )+b^6 x^6 \left (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\right )\right )}{12012 (a+b x)} \end {gather*}
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))
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IntegrateAlgebraic [F] time = 5.48, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
Defer[IntegrateAlgebraic][(a + b*x)*(d + e*x)^6*(a^2 + 2*a*b*x + b^2*x^2)^(5/2), x]
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fricas [B] time = 0.43, size = 599, normalized size = 1.93 \begin {gather*} \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 {gather*}
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
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giac [B] time = 0.21, size = 955, normalized size = 3.07
result too large to display
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)
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maple [B] time = 0.05, size = 707, normalized size = 2.27 \begin {gather*} \frac {\left (924 b^{6} e^{6} x^{12}+6006 x^{11} a \,b^{5} e^{6}+6006 x^{11} b^{6} d \,e^{5}+16380 x^{10} a^{2} b^{4} e^{6}+39312 x^{10} a \,b^{5} d \,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} a \,b^{5} 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} a \,b^{5} d^{4} e^{2}+9009 x^{7} b^{6} d^{5} e +1716 x^{6} a^{6} e^{6}+61776 x^{6} a^{5} b d \,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} a \,b^{5} 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} e x +36036 a^{5} b \,d^{6} x +12012 a^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{12012 \left (b x +a \right )^{5}} \end {gather*}
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
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maxima [B] time = 0.70, size = 1736, normalized size = 5.58
result too large to display
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]
1/13*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^6*x^6/b - 19/156*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^6*x^5/b^2 + 251/17
16*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^6*x^4/b^3 - 68/429*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*e^6*x^3/b^4 +
1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^6*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7*e^6*x/b^6 + 211/1287*(b^
2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*e^6*x^2/b^5 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*d^6/b - 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*a^8*e^6/b^7 - 1709/10296*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5*e^6*x/b^6 + 1715/10296*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*a^6*e^6/b^7 + 1/12*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^5/b^2 - 17
/132*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^4/b^3 + 3/11*(5*b*d^2*e^4 + 2*a*d*e^5)*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*x^4/b^2 + 5/33*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^3/b^4 - 9/22*(5
*b*d^2*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x^3/b^3 + 1/2*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*x^3/b^2 + 1/6*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^6*x/b^6 - 1/2*(5*b*d
^2*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*x/b^5 + 5/6*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a
*b*x + a^2)^(5/2)*a^4*x/b^4 - 5/6*(3*b*d^4*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 1/2*
(2*b*d^5*e + 5*a*d^4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(b*d^6 + 6*a*d^5*e)*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*a*x/b - 16/99*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^3*x^2/b^5 + 31/66*(5*b*d^2
*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x^2/b^4 - 13/18*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2
*a*b*x + a^2)^(7/2)*a*x^2/b^3 + 5/9*(3*b*d^4*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x^2/b^2 + 1/6*
(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^7/b^7 - 1/2*(5*b*d^2*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a^6/b^6 + 5/6*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5/b^5 - 5/6*(3*b*d^4
*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 1/2*(2*b*d^5*e + 5*a*d^4*e^2)*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a^3/b^3 - 1/6*(b*d^6 + 6*a*d^5*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 + 131/792*(6*b*d*e^5 +
a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4*x/b^6 - 65/132*(5*b*d^2*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)
^(7/2)*a^3*x/b^5 + 29/36*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*x/b^4 - 55/72*(3*b*d^
4*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 3/8*(2*b*d^5*e + 5*a*d^4*e^2)*(b^2*x^2 + 2*a*b*
x + a^2)^(7/2)*x/b^2 - 923/5544*(6*b*d*e^5 + a*e^6)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^5/b^7 + 461/924*(5*b*d^2
*e^4 + 2*a*d*e^5)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^4/b^6 - 209/252*(4*b*d^3*e^3 + 3*a*d^2*e^4)*(b^2*x^2 + 2*a
*b*x + a^2)^(7/2)*a^3/b^5 + 415/504*(3*b*d^4*e^2 + 4*a*d^3*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 27/5
6*(2*b*d^5*e + 5*a*d^4*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(b*d^6 + 6*a*d^5*e)*(b^2*x^2 + 2*a*b*x
+ a^2)^(7/2)/b^2
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^6\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
[Out]
int((a + b*x)*(d + e*x)^6*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right )^{6} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
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]
Integral((a + b*x)*(d + e*x)**6*((a + b*x)**2)**(5/2), x)
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