3.1496 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=167 \[ \frac {6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {b^6 \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac {6 b^5 (b d-a e)}{e^7 (d+e x)}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {b^6 \log (d+e x)}{e^7} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^7,x]

[Out]

-(b*d - a*e)^6/(6*e^7*(d + e*x)^6) + (6*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^5) - (15*b^2*(b*d - a*e)^4)/(4*e^7*(
d + e*x)^4) + (20*b^3*(b*d - a*e)^3)/(3*e^7*(d + e*x)^3) - (15*b^4*(b*d - a*e)^2)/(2*e^7*(d + e*x)^2) + (6*b^5
*(b*d - a*e))/(e^7*(d + e*x)) + (b^6*Log[d + e*x])/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 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 \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^7} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^7} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^7}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac {b^6}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {(b d-a e)^6}{6 e^7 (d+e x)^6}+\frac {6 b (b d-a e)^5}{5 e^7 (d+e x)^5}-\frac {15 b^2 (b d-a e)^4}{4 e^7 (d+e x)^4}+\frac {20 b^3 (b d-a e)^3}{3 e^7 (d+e x)^3}-\frac {15 b^4 (b d-a e)^2}{2 e^7 (d+e x)^2}+\frac {6 b^5 (b d-a e)}{e^7 (d+e x)}+\frac {b^6 \log (d+e x)}{e^7}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 233, normalized size = 1.40 \[ \frac {\frac {(b d-a e) \left (10 a^5 e^5+2 a^4 b e^4 (11 d+36 e x)+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+a^2 b^3 e^2 \left (57 d^3+282 d^2 e x+525 d e^2 x^2+400 e^3 x^3\right )+a b^4 e \left (87 d^4+462 d^3 e x+975 d^2 e^2 x^2+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )}{(d+e x)^6}+60 b^6 \log (d+e x)}{60 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^7,x]

[Out]

(((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x + 225*e^2*x^2) + a^2
*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*x + 975*d^2*e^2*x^
2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*
e^4*x^4 + 360*e^5*x^5)))/(d + e*x)^6 + 60*b^6*Log[d + e*x])/(60*e^7)

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fricas [B]  time = 0.95, size = 492, normalized size = 2.95 \[ \frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \, {\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*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 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + 6*b^6*d*e^5*x^5 +
15*b^6*d^2*e^4*x^4 + 20*b^6*d^3*e^3*x^3 + 15*b^6*d^4*e^2*x^2 + 6*b^6*d^5*e*x + b^6*d^6)*log(e*x + d))/(e^13*x
^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)

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giac [B]  time = 0.17, size = 339, normalized size = 2.03 \[ b^{6} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (360 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{3} - 2 \, a b^{5} d e^{4} - a^{2} b^{4} e^{5}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{2} - 6 \, a b^{5} d^{2} e^{3} - 3 \, a^{2} b^{4} d e^{4} - 2 \, a^{3} b^{3} e^{5}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e - 12 \, a b^{5} d^{3} e^{2} - 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} - 3 \, a^{4} b^{2} e^{5}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} - 60 \, a b^{5} d^{4} e - 30 \, a^{2} b^{4} d^{3} e^{2} - 20 \, a^{3} b^{3} d^{2} e^{3} - 15 \, a^{4} b^{2} d e^{4} - 12 \, a^{5} b e^{5}\right )} x + {\left (147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^7,x, algorithm="giac")

[Out]

b^6*e^(-7)*log(abs(x*e + d)) + 1/60*(360*(b^6*d*e^4 - a*b^5*e^5)*x^5 + 450*(3*b^6*d^2*e^3 - 2*a*b^5*d*e^4 - a^
2*b^4*e^5)*x^4 + 200*(11*b^6*d^3*e^2 - 6*a*b^5*d^2*e^3 - 3*a^2*b^4*d*e^4 - 2*a^3*b^3*e^5)*x^3 + 75*(25*b^6*d^4
*e - 12*a*b^5*d^3*e^2 - 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 - 3*a^4*b^2*e^5)*x^2 + 6*(137*b^6*d^5 - 60*a*b^5*d
^4*e - 30*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 15*a^4*b^2*d*e^4 - 12*a^5*b*e^5)*x + (147*b^6*d^6 - 60*a*b^5*
d^5*e - 30*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 - 12*a^5*b*d*e^5 - 10*a^6*e^6)*e^(-1))*e^
(-6)/(x*e + d)^6

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maple [B]  time = 0.05, size = 513, normalized size = 3.07 \[ -\frac {a^{6}}{6 \left (e x +d \right )^{6} e}+\frac {a^{5} b d}{\left (e x +d \right )^{6} e^{2}}-\frac {5 a^{4} b^{2} d^{2}}{2 \left (e x +d \right )^{6} e^{3}}+\frac {10 a^{3} b^{3} d^{3}}{3 \left (e x +d \right )^{6} e^{4}}-\frac {5 a^{2} b^{4} d^{4}}{2 \left (e x +d \right )^{6} e^{5}}+\frac {a \,b^{5} d^{5}}{\left (e x +d \right )^{6} e^{6}}-\frac {b^{6} d^{6}}{6 \left (e x +d \right )^{6} e^{7}}-\frac {6 a^{5} b}{5 \left (e x +d \right )^{5} e^{2}}+\frac {6 a^{4} b^{2} d}{\left (e x +d \right )^{5} e^{3}}-\frac {12 a^{3} b^{3} d^{2}}{\left (e x +d \right )^{5} e^{4}}+\frac {12 a^{2} b^{4} d^{3}}{\left (e x +d \right )^{5} e^{5}}-\frac {6 a \,b^{5} d^{4}}{\left (e x +d \right )^{5} e^{6}}+\frac {6 b^{6} d^{5}}{5 \left (e x +d \right )^{5} e^{7}}-\frac {15 a^{4} b^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {15 a^{3} b^{3} d}{\left (e x +d \right )^{4} e^{4}}-\frac {45 a^{2} b^{4} d^{2}}{2 \left (e x +d \right )^{4} e^{5}}+\frac {15 a \,b^{5} d^{3}}{\left (e x +d \right )^{4} e^{6}}-\frac {15 b^{6} d^{4}}{4 \left (e x +d \right )^{4} e^{7}}-\frac {20 a^{3} b^{3}}{3 \left (e x +d \right )^{3} e^{4}}+\frac {20 a^{2} b^{4} d}{\left (e x +d \right )^{3} e^{5}}-\frac {20 a \,b^{5} d^{2}}{\left (e x +d \right )^{3} e^{6}}+\frac {20 b^{6} d^{3}}{3 \left (e x +d \right )^{3} e^{7}}-\frac {15 a^{2} b^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {15 a \,b^{5} d}{\left (e x +d \right )^{2} e^{6}}-\frac {15 b^{6} d^{2}}{2 \left (e x +d \right )^{2} e^{7}}-\frac {6 a \,b^{5}}{\left (e x +d \right ) e^{6}}+\frac {6 b^{6} d}{\left (e x +d \right ) e^{7}}+\frac {b^{6} \ln \left (e x +d \right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^7,x)

[Out]

-1/6/e^7/(e*x+d)^6*b^6*d^6-20/3*b^3/e^4/(e*x+d)^3*a^3+20/3*b^6/e^7/(e*x+d)^3*d^3-15/4*b^2/e^3/(e*x+d)^4*a^4-15
/4*b^6/e^7/(e*x+d)^4*d^4-15/2*b^4/e^5/(e*x+d)^2*a^2-15/2*b^6/e^7/(e*x+d)^2*d^2-6*b^5/e^6/(e*x+d)*a+6*b^6/e^7/(
e*x+d)*d+6/5*b^6/e^7/(e*x+d)^5*d^5-6/5*b/e^2/(e*x+d)^5*a^5+12*b^4/e^5/(e*x+d)^5*a^2*d^3-6*b^5/e^6/(e*x+d)^5*a*
d^4+15*b^3/e^4/(e*x+d)^4*d*a^3-1/6/e/(e*x+d)^6*a^6-45/2*b^4/e^5/(e*x+d)^4*a^2*d^2+15*b^5/e^6/(e*x+d)^4*a*d^3+1
5*b^5/e^6/(e*x+d)^2*a*d+1/e^2/(e*x+d)^6*d*a^5*b+20*b^4/e^5/(e*x+d)^3*a^2*d-20*b^5/e^6/(e*x+d)^3*a*d^2+6*b^2/e^
3/(e*x+d)^5*a^4*d-12*b^3/e^4/(e*x+d)^5*a^3*d^2-5/2/e^3/(e*x+d)^6*d^2*a^4*b^2+10/3/e^4/(e*x+d)^6*d^3*a^3*b^3-5/
2/e^5/(e*x+d)^6*d^4*a^2*b^4+1/e^6/(e*x+d)^6*d^5*a*b^5+b^6*ln(e*x+d)/e^7

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maxima [B]  time = 1.54, size = 416, normalized size = 2.49 \[ \frac {147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \, {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \, {\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \, {\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \, {\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \, {\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {b^{6} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*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 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^
11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + b^6*log(e*x + d)/e^7

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mupad [B]  time = 0.64, size = 353, normalized size = 2.11 \[ \frac {b^6\,\ln \left (d+e\,x\right )}{e^7}-\frac {x^5\,\left (6\,a\,b^5\,e^6-6\,b^6\,d\,e^5\right )+x^2\,\left (\frac {15\,a^4\,b^2\,e^6}{4}+5\,a^3\,b^3\,d\,e^5+\frac {15\,a^2\,b^4\,d^2\,e^4}{2}+15\,a\,b^5\,d^3\,e^3-\frac {125\,b^6\,d^4\,e^2}{4}\right )+x^4\,\left (\frac {15\,a^2\,b^4\,e^6}{2}+15\,a\,b^5\,d\,e^5-\frac {45\,b^6\,d^2\,e^4}{2}\right )+x\,\left (\frac {6\,a^5\,b\,e^6}{5}+\frac {3\,a^4\,b^2\,d\,e^5}{2}+2\,a^3\,b^3\,d^2\,e^4+3\,a^2\,b^4\,d^3\,e^3+6\,a\,b^5\,d^4\,e^2-\frac {137\,b^6\,d^5\,e}{10}\right )+\frac {a^6\,e^6}{6}-\frac {49\,b^6\,d^6}{20}+x^3\,\left (\frac {20\,a^3\,b^3\,e^6}{3}+10\,a^2\,b^4\,d\,e^5+20\,a\,b^5\,d^2\,e^4-\frac {110\,b^6\,d^3\,e^3}{3}\right )+\frac {a^2\,b^4\,d^4\,e^2}{2}+\frac {a^3\,b^3\,d^3\,e^3}{3}+\frac {a^4\,b^2\,d^2\,e^4}{4}+a\,b^5\,d^5\,e+\frac {a^5\,b\,d\,e^5}{5}}{e^7\,{\left (d+e\,x\right )}^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^7,x)

[Out]

(b^6*log(d + e*x))/e^7 - (x^5*(6*a*b^5*e^6 - 6*b^6*d*e^5) + x^2*((15*a^4*b^2*e^6)/4 - (125*b^6*d^4*e^2)/4 + 15
*a*b^5*d^3*e^3 + 5*a^3*b^3*d*e^5 + (15*a^2*b^4*d^2*e^4)/2) + x^4*((15*a^2*b^4*e^6)/2 - (45*b^6*d^2*e^4)/2 + 15
*a*b^5*d*e^5) + x*((6*a^5*b*e^6)/5 - (137*b^6*d^5*e)/10 + 6*a*b^5*d^4*e^2 + (3*a^4*b^2*d*e^5)/2 + 3*a^2*b^4*d^
3*e^3 + 2*a^3*b^3*d^2*e^4) + (a^6*e^6)/6 - (49*b^6*d^6)/20 + x^3*((20*a^3*b^3*e^6)/3 - (110*b^6*d^3*e^3)/3 + 2
0*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5) + (a^2*b^4*d^4*e^2)/2 + (a^3*b^3*d^3*e^3)/3 + (a^4*b^2*d^2*e^4)/4 + a*b^5*
d^5*e + (a^5*b*d*e^5)/5)/(e^7*(d + e*x)^6)

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sympy [B]  time = 99.03, size = 439, normalized size = 2.63 \[ \frac {b^{6} \log {\left (d + e x \right )}}{e^{7}} + \frac {- 10 a^{6} e^{6} - 12 a^{5} b d e^{5} - 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} - 30 a^{2} b^{4} d^{4} e^{2} - 60 a b^{5} d^{5} e + 147 b^{6} d^{6} + x^{5} \left (- 360 a b^{5} e^{6} + 360 b^{6} d e^{5}\right ) + x^{4} \left (- 450 a^{2} b^{4} e^{6} - 900 a b^{5} d e^{5} + 1350 b^{6} d^{2} e^{4}\right ) + x^{3} \left (- 400 a^{3} b^{3} e^{6} - 600 a^{2} b^{4} d e^{5} - 1200 a b^{5} d^{2} e^{4} + 2200 b^{6} d^{3} e^{3}\right ) + x^{2} \left (- 225 a^{4} b^{2} e^{6} - 300 a^{3} b^{3} d e^{5} - 450 a^{2} b^{4} d^{2} e^{4} - 900 a b^{5} d^{3} e^{3} + 1875 b^{6} d^{4} e^{2}\right ) + x \left (- 72 a^{5} b e^{6} - 90 a^{4} b^{2} d e^{5} - 120 a^{3} b^{3} d^{2} e^{4} - 180 a^{2} b^{4} d^{3} e^{3} - 360 a b^{5} d^{4} e^{2} + 822 b^{6} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**7,x)

[Out]

b**6*log(d + e*x)/e**7 + (-10*a**6*e**6 - 12*a**5*b*d*e**5 - 15*a**4*b**2*d**2*e**4 - 20*a**3*b**3*d**3*e**3 -
30*a**2*b**4*d**4*e**2 - 60*a*b**5*d**5*e + 147*b**6*d**6 + x**5*(-360*a*b**5*e**6 + 360*b**6*d*e**5) + x**4*
(-450*a**2*b**4*e**6 - 900*a*b**5*d*e**5 + 1350*b**6*d**2*e**4) + x**3*(-400*a**3*b**3*e**6 - 600*a**2*b**4*d*
e**5 - 1200*a*b**5*d**2*e**4 + 2200*b**6*d**3*e**3) + x**2*(-225*a**4*b**2*e**6 - 300*a**3*b**3*d*e**5 - 450*a
**2*b**4*d**2*e**4 - 900*a*b**5*d**3*e**3 + 1875*b**6*d**4*e**2) + x*(-72*a**5*b*e**6 - 90*a**4*b**2*d*e**5 -
120*a**3*b**3*d**2*e**4 - 180*a**2*b**4*d**3*e**3 - 360*a*b**5*d**4*e**2 + 822*b**6*d**5*e))/(60*d**6*e**7 + 3
60*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13
*x**6)

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