3.21.9 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{12}} \, dx\) [2009]

Optimal. Leaf size=359 \[ -\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {3 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {5 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^9}+\frac {5 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^8}-\frac {15 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac {b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^6}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5} \]

[Out]

-1/11*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^11+3/5*b*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(
e*x+d)^10-5/3*b^2*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^9+5/2*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2)/
e^7/(b*x+a)/(e*x+d)^8-15/7*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^7+b^5*(-a*e+b*d)*((b*x+a)^2)
^(1/2)/e^7/(b*x+a)/(e*x+d)^6-1/5*b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^5

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Rubi [A]
time = 0.14, antiderivative size = 359, 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 = {784, 21, 45} \begin {gather*} -\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^7 (a+b x) (d+e x)^9}+\frac {3 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)^6}-\frac {15 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^7 (a+b x) (d+e x)^7}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{2 e^7 (a+b x) (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]

[Out]

-1/11*((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^11) + (3*b*(b*d - a*e)^5*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^10) - (5*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3
*e^7*(a + b*x)*(d + e*x)^9) + (5*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^8
) - (15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^7) + (b^5*(b*d - a*e)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^6) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*
(d + e*x)^5)

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 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 784

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

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Mathematica [A]
time = 0.07, size = 295, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )}{2310 e^7 (a+b x) (d+e x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]

[Out]

-1/2310*(Sqrt[(a + b*x)^2]*(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70*a^4*b^2*e^4*(d^2 + 11*d*e*x + 55*e^2
*x^2) + 35*a^3*b^3*e^3*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 15*a^2*b^4*e^2*(d^4 + 11*d^3*e*x + 55
*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 +
330*d*e^4*x^4 + 462*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 4
62*d*e^5*x^5 + 462*e^6*x^6)))/(e^7*(a + b*x)*(d + e*x)^11)

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Maple [A]
time = 0.08, size = 392, normalized size = 1.09

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{6} x^{6}}{5 e}-\frac {b^{5} \left (5 a e +b d \right ) x^{5}}{5 e^{2}}-\frac {b^{4} \left (15 a^{2} e^{2}+5 a b d e +b^{2} d^{2}\right ) x^{4}}{7 e^{3}}-\frac {b^{3} \left (35 a^{3} e^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{14 e^{4}}-\frac {b^{2} \left (70 a^{4} e^{4}+35 a^{3} b d \,e^{3}+15 a^{2} b^{2} d^{2} e^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{42 e^{5}}-\frac {b \left (126 a^{5} e^{5}+70 a^{4} b d \,e^{4}+35 a^{3} b^{2} d^{2} e^{3}+15 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{210 e^{6}}-\frac {210 e^{6} a^{6}+126 d \,e^{5} a^{5} b +70 d^{2} e^{4} a^{4} b^{2}+35 d^{3} e^{3} a^{3} b^{3}+15 d^{4} e^{2} a^{2} b^{4}+5 d^{5} e a \,b^{5}+d^{6} b^{6}}{2310 e^{7}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{11}}\)	\(351\)
gosper	\(-\frac {\left (462 b^{6} e^{6} x^{6}+2310 a \,b^{5} e^{6} x^{5}+462 b^{6} d \,e^{5} x^{5}+4950 a^{2} b^{4} e^{6} x^{4}+1650 a \,b^{5} d \,e^{5} x^{4}+330 b^{6} d^{2} e^{4} x^{4}+5775 a^{3} b^{3} e^{6} x^{3}+2475 a^{2} b^{4} d \,e^{5} x^{3}+825 a \,b^{5} d^{2} e^{4} x^{3}+165 b^{6} d^{3} e^{3} x^{3}+3850 a^{4} b^{2} e^{6} x^{2}+1925 a^{3} b^{3} d \,e^{5} x^{2}+825 a^{2} b^{4} d^{2} e^{4} x^{2}+275 a \,b^{5} d^{3} e^{3} x^{2}+55 b^{6} d^{4} e^{2} x^{2}+1386 a^{5} b \,e^{6} x +770 a^{4} b^{2} d \,e^{5} x +385 a^{3} b^{3} d^{2} e^{4} x +165 a^{2} b^{4} d^{3} e^{3} x +55 a \,b^{5} d^{4} e^{2} x +11 b^{6} d^{5} e x +210 e^{6} a^{6}+126 d \,e^{5} a^{5} b +70 d^{2} e^{4} a^{4} b^{2}+35 d^{3} e^{3} a^{3} b^{3}+15 d^{4} e^{2} a^{2} b^{4}+5 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2310 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\)	\(392\)
default	\(-\frac {\left (462 b^{6} e^{6} x^{6}+2310 a \,b^{5} e^{6} x^{5}+462 b^{6} d \,e^{5} x^{5}+4950 a^{2} b^{4} e^{6} x^{4}+1650 a \,b^{5} d \,e^{5} x^{4}+330 b^{6} d^{2} e^{4} x^{4}+5775 a^{3} b^{3} e^{6} x^{3}+2475 a^{2} b^{4} d \,e^{5} x^{3}+825 a \,b^{5} d^{2} e^{4} x^{3}+165 b^{6} d^{3} e^{3} x^{3}+3850 a^{4} b^{2} e^{6} x^{2}+1925 a^{3} b^{3} d \,e^{5} x^{2}+825 a^{2} b^{4} d^{2} e^{4} x^{2}+275 a \,b^{5} d^{3} e^{3} x^{2}+55 b^{6} d^{4} e^{2} x^{2}+1386 a^{5} b \,e^{6} x +770 a^{4} b^{2} d \,e^{5} x +385 a^{3} b^{3} d^{2} e^{4} x +165 a^{2} b^{4} d^{3} e^{3} x +55 a \,b^{5} d^{4} e^{2} x +11 b^{6} d^{5} e x +210 e^{6} a^{6}+126 d \,e^{5} a^{5} b +70 d^{2} e^{4} a^{4} b^{2}+35 d^{3} e^{3} a^{3} b^{3}+15 d^{4} e^{2} a^{2} b^{4}+5 d^{5} e a \,b^{5}+d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2310 e^{7} \left (e x +d \right )^{11} \left (b x +a \right )^{5}}\)	\(392\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x,method=_RETURNVERBOSE)

[Out]

-1/2310/e^7*(462*b^6*e^6*x^6+2310*a*b^5*e^6*x^5+462*b^6*d*e^5*x^5+4950*a^2*b^4*e^6*x^4+1650*a*b^5*d*e^5*x^4+33
0*b^6*d^2*e^4*x^4+5775*a^3*b^3*e^6*x^3+2475*a^2*b^4*d*e^5*x^3+825*a*b^5*d^2*e^4*x^3+165*b^6*d^3*e^3*x^3+3850*a
^4*b^2*e^6*x^2+1925*a^3*b^3*d*e^5*x^2+825*a^2*b^4*d^2*e^4*x^2+275*a*b^5*d^3*e^3*x^2+55*b^6*d^4*e^2*x^2+1386*a^
5*b*e^6*x+770*a^4*b^2*d*e^5*x+385*a^3*b^3*d^2*e^4*x+165*a^2*b^4*d^3*e^3*x+55*a*b^5*d^4*e^2*x+11*b^6*d^5*e*x+21
0*a^6*e^6+126*a^5*b*d*e^5+70*a^4*b^2*d^2*e^4+35*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2+5*a*b^5*d^5*e+b^6*d^6)*((b*
x+a)^2)^(5/2)/(e*x+d)^11/(b*x+a)^5

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [A]
time = 2.89, size = 426, normalized size = 1.19 \begin {gather*} -\frac {b^{6} d^{6} + {\left (462 \, b^{6} x^{6} + 2310 \, a b^{5} x^{5} + 4950 \, a^{2} b^{4} x^{4} + 5775 \, a^{3} b^{3} x^{3} + 3850 \, a^{4} b^{2} x^{2} + 1386 \, a^{5} b x + 210 \, a^{6}\right )} e^{6} + {\left (462 \, b^{6} d x^{5} + 1650 \, a b^{5} d x^{4} + 2475 \, a^{2} b^{4} d x^{3} + 1925 \, a^{3} b^{3} d x^{2} + 770 \, a^{4} b^{2} d x + 126 \, a^{5} b d\right )} e^{5} + 5 \, {\left (66 \, b^{6} d^{2} x^{4} + 165 \, a b^{5} d^{2} x^{3} + 165 \, a^{2} b^{4} d^{2} x^{2} + 77 \, a^{3} b^{3} d^{2} x + 14 \, a^{4} b^{2} d^{2}\right )} e^{4} + 5 \, {\left (33 \, b^{6} d^{3} x^{3} + 55 \, a b^{5} d^{3} x^{2} + 33 \, a^{2} b^{4} d^{3} x + 7 \, a^{3} b^{3} d^{3}\right )} e^{3} + 5 \, {\left (11 \, b^{6} d^{4} x^{2} + 11 \, a b^{5} d^{4} x + 3 \, a^{2} b^{4} d^{4}\right )} e^{2} + {\left (11 \, b^{6} d^{5} x + 5 \, a b^{5} d^{5}\right )} e}{2310 \, {\left (x^{11} e^{18} + 11 \, d x^{10} e^{17} + 55 \, d^{2} x^{9} e^{16} + 165 \, d^{3} x^{8} e^{15} + 330 \, d^{4} x^{7} e^{14} + 462 \, d^{5} x^{6} e^{13} + 462 \, d^{6} x^{5} e^{12} + 330 \, d^{7} x^{4} e^{11} + 165 \, d^{8} x^{3} e^{10} + 55 \, d^{9} x^{2} e^{9} + 11 \, d^{10} x e^{8} + d^{11} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="fricas")

[Out]

-1/2310*(b^6*d^6 + (462*b^6*x^6 + 2310*a*b^5*x^5 + 4950*a^2*b^4*x^4 + 5775*a^3*b^3*x^3 + 3850*a^4*b^2*x^2 + 13
86*a^5*b*x + 210*a^6)*e^6 + (462*b^6*d*x^5 + 1650*a*b^5*d*x^4 + 2475*a^2*b^4*d*x^3 + 1925*a^3*b^3*d*x^2 + 770*
a^4*b^2*d*x + 126*a^5*b*d)*e^5 + 5*(66*b^6*d^2*x^4 + 165*a*b^5*d^2*x^3 + 165*a^2*b^4*d^2*x^2 + 77*a^3*b^3*d^2*
x + 14*a^4*b^2*d^2)*e^4 + 5*(33*b^6*d^3*x^3 + 55*a*b^5*d^3*x^2 + 33*a^2*b^4*d^3*x + 7*a^3*b^3*d^3)*e^3 + 5*(11
*b^6*d^4*x^2 + 11*a*b^5*d^4*x + 3*a^2*b^4*d^4)*e^2 + (11*b^6*d^5*x + 5*a*b^5*d^5)*e)/(x^11*e^18 + 11*d*x^10*e^
17 + 55*d^2*x^9*e^16 + 165*d^3*x^8*e^15 + 330*d^4*x^7*e^14 + 462*d^5*x^6*e^13 + 462*d^6*x^5*e^12 + 330*d^7*x^4
*e^11 + 165*d^8*x^3*e^10 + 55*d^9*x^2*e^9 + 11*d^10*x*e^8 + d^11*e^7)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**12,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [A]
time = 1.21, size = 520, normalized size = 1.45 \begin {gather*} -\frac {{\left (462 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 462 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 330 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 11 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 2310 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1650 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 825 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 275 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 55 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 4950 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 2475 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 825 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 165 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 5775 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 1925 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 385 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 3850 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 770 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1386 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + 126 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{2310 \, {\left (x e + d\right )}^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="giac")

[Out]

-1/2310*(462*b^6*x^6*e^6*sgn(b*x + a) + 462*b^6*d*x^5*e^5*sgn(b*x + a) + 330*b^6*d^2*x^4*e^4*sgn(b*x + a) + 16
5*b^6*d^3*x^3*e^3*sgn(b*x + a) + 55*b^6*d^4*x^2*e^2*sgn(b*x + a) + 11*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sgn(b
*x + a) + 2310*a*b^5*x^5*e^6*sgn(b*x + a) + 1650*a*b^5*d*x^4*e^5*sgn(b*x + a) + 825*a*b^5*d^2*x^3*e^4*sgn(b*x
+ a) + 275*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 55*a*b^5*d^4*x*e^2*sgn(b*x + a) + 5*a*b^5*d^5*e*sgn(b*x + a) + 495
0*a^2*b^4*x^4*e^6*sgn(b*x + a) + 2475*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 825*a^2*b^4*d^2*x^2*e^4*sgn(b*x + a) +
165*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) + 5775*a^3*b^3*x^3*e^6*sgn(b*x + a) + 192
5*a^3*b^3*d*x^2*e^5*sgn(b*x + a) + 385*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 35*a^3*b^3*d^3*e^3*sgn(b*x + a) + 3850
*a^4*b^2*x^2*e^6*sgn(b*x + a) + 770*a^4*b^2*d*x*e^5*sgn(b*x + a) + 70*a^4*b^2*d^2*e^4*sgn(b*x + a) + 1386*a^5*
b*x*e^6*sgn(b*x + a) + 126*a^5*b*d*e^5*sgn(b*x + a) + 210*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^11

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Mupad [B]
time = 2.53, size = 1010, normalized size = 2.81 \begin {gather*} \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{10\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{10\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{10\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{10\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{10\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{10\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{7\,e^7}+\frac {d\,\left (\frac {b^6\,d}{7\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{7\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {a^6}{11\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{11\,e}-\frac {b^6\,d}{11\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{11\,e}\right )}{e}+\frac {20\,a^3\,b^3}{11\,e}\right )}{e}-\frac {15\,a^4\,b^2}{11\,e}\right )}{e}+\frac {6\,a^5\,b}{11\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{9\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{9\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{9\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{9\,e^4}\right )}{e}+\frac {b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{3\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{6\,e^7}+\frac {b^6\,d}{6\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{8\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{8\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{8\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{8\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^12,x)

[Out]

(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(10*e^7
) + (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6*a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(10*e^7) - (d*(
(20*a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(10*e^7) - (d*((d*((b^6*d)/(10*e^3) - (b^5
*(6*a*e - b*d))/(10*e^3)))/e + (b^4*(15*a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(10*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 +
2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5*d*e)/(7*e^7) + (d*((b^6*
d)/(7*e^6) - (2*b^5*(3*a*e - 2*b*d))/(7*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (
(a^6/(11*e) - (d*((d*((d*((d*((d*((6*a*b^5)/(11*e) - (b^6*d)/(11*e^2)))/e - (15*a^2*b^4)/(11*e)))/e + (20*a^3*
b^3)/(11*e)))/e - (15*a^4*b^2)/(11*e)))/e + (6*a^5*b)/(11*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*
(d + e*x)^11) - (((5*b^6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d^3*e)/(9*e^7
) + (d*((4*b^6*d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30*a^2*b^4*d*e^3)/(9*e^7) + (d*((d*((b^6*d)/(9*e^4)
- (2*b^5*(3*a*e - b*d))/(9*e^4)))/e + (b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(3*e^5)))/e))/e)*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) + (((5*b^6*d - 6*a*b^5*e)/(6*e^7) + (b^6*d)/(6*e^7))*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d*e^2 - 36*a*b^5*d^2*
e)/(8*e^7) + (d*((d*((b^6*d)/(8*e^5) - (3*b^5*(2*a*e - b*d))/(8*e^5)))/e + (3*b^4*(5*a^2*e^2 + 2*b^2*d^2 - 6*a
*b*d*e))/(8*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^8) - (b^6*(a^2 + b^2*x^2 + 2*a*b*x
)^(1/2))/(5*e^7*(a + b*x)*(d + e*x)^5)

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