3.269 \(\int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx\)

Optimal. Leaf size=200 \[ -\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}+\frac {3 a^2 x (b c-2 a d) (b c-a d)^2}{b^7}-\frac {a x^2 (2 b c-5 a d) (b c-a d)^2}{2 b^6}+\frac {x^3 (b c-4 a d) (b c-a d)^2}{3 b^5}+\frac {3 d x^4 (b c-a d)^2}{4 b^4}+\frac {d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac {d^3 x^6}{6 b^2} \]

[Out]

3*a^2*(-2*a*d+b*c)*(-a*d+b*c)^2*x/b^7-1/2*a*(-5*a*d+2*b*c)*(-a*d+b*c)^2*x^2/b^6+1/3*(-4*a*d+b*c)*(-a*d+b*c)^2*
x^3/b^5+3/4*d*(-a*d+b*c)^2*x^4/b^4+1/5*d^2*(-2*a*d+3*b*c)*x^5/b^3+1/6*d^3*x^6/b^2-a^4*(-a*d+b*c)^3/b^8/(b*x+a)
-a^3*(-7*a*d+4*b*c)*(-a*d+b*c)^2*ln(b*x+a)/b^8

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Rubi [A]  time = 0.27, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ \frac {3 a^2 x (b c-2 a d) (b c-a d)^2}{b^7}-\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}+\frac {d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac {3 d x^4 (b c-a d)^2}{4 b^4}+\frac {x^3 (b c-4 a d) (b c-a d)^2}{3 b^5}-\frac {a x^2 (2 b c-5 a d) (b c-a d)^2}{2 b^6}+\frac {d^3 x^6}{6 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

(3*a^2*(b*c - 2*a*d)*(b*c - a*d)^2*x)/b^7 - (a*(2*b*c - 5*a*d)*(b*c - a*d)^2*x^2)/(2*b^6) + ((b*c - 4*a*d)*(b*
c - a*d)^2*x^3)/(3*b^5) + (3*d*(b*c - a*d)^2*x^4)/(4*b^4) + (d^2*(3*b*c - 2*a*d)*x^5)/(5*b^3) + (d^3*x^6)/(6*b
^2) - (a^4*(b*c - a*d)^3)/(b^8*(a + b*x)) - (a^3*(4*b*c - 7*a*d)*(b*c - a*d)^2*Log[a + b*x])/b^8

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx &=\int \left (-\frac {3 a^2 (-b c+a d)^2 (-b c+2 a d)}{b^7}+\frac {a (-b c+a d)^2 (-2 b c+5 a d) x}{b^6}+\frac {(b c-4 a d) (b c-a d)^2 x^2}{b^5}+\frac {3 d (b c-a d)^2 x^3}{b^4}+\frac {d^2 (3 b c-2 a d) x^4}{b^3}+\frac {d^3 x^5}{b^2}-\frac {a^4 (-b c+a d)^3}{b^7 (a+b x)^2}+\frac {a^3 (-b c+a d)^2 (-4 b c+7 a d)}{b^7 (a+b x)}\right ) \, dx\\ &=\frac {3 a^2 (b c-2 a d) (b c-a d)^2 x}{b^7}-\frac {a (2 b c-5 a d) (b c-a d)^2 x^2}{2 b^6}+\frac {(b c-4 a d) (b c-a d)^2 x^3}{3 b^5}+\frac {3 d (b c-a d)^2 x^4}{4 b^4}+\frac {d^2 (3 b c-2 a d) x^5}{5 b^3}+\frac {d^3 x^6}{6 b^2}-\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 190, normalized size = 0.95 \[ \frac {\frac {60 a^4 (a d-b c)^3}{a+b x}+60 a^3 (b c-a d)^2 (7 a d-4 b c) \log (a+b x)-180 a^2 b x (b c-a d)^2 (2 a d-b c)+12 b^5 d^2 x^5 (3 b c-2 a d)+45 b^4 d x^4 (b c-a d)^2+20 b^3 x^3 (b c-4 a d) (b c-a d)^2+30 a b^2 x^2 (b c-a d)^2 (5 a d-2 b c)+10 b^6 d^3 x^6}{60 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x)^3)/(a + b*x)^2,x]

[Out]

(-180*a^2*b*(b*c - a*d)^2*(-(b*c) + 2*a*d)*x + 30*a*b^2*(b*c - a*d)^2*(-2*b*c + 5*a*d)*x^2 + 20*b^3*(b*c - 4*a
*d)*(b*c - a*d)^2*x^3 + 45*b^4*d*(b*c - a*d)^2*x^4 + 12*b^5*d^2*(3*b*c - 2*a*d)*x^5 + 10*b^6*d^3*x^6 + (60*a^4
*(-(b*c) + a*d)^3)/(a + b*x) + 60*a^3*(b*c - a*d)^2*(-4*b*c + 7*a*d)*Log[a + b*x])/(60*b^8)

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fricas [B]  time = 0.94, size = 421, normalized size = 2.10 \[ \frac {10 \, b^{7} d^{3} x^{7} - 60 \, a^{4} b^{3} c^{3} + 180 \, a^{5} b^{2} c^{2} d - 180 \, a^{6} b c d^{2} + 60 \, a^{7} d^{3} + 2 \, {\left (18 \, b^{7} c d^{2} - 7 \, a b^{6} d^{3}\right )} x^{6} + 3 \, {\left (15 \, b^{7} c^{2} d - 18 \, a b^{6} c d^{2} + 7 \, a^{2} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (4 \, b^{7} c^{3} - 15 \, a b^{6} c^{2} d + 18 \, a^{2} b^{5} c d^{2} - 7 \, a^{3} b^{4} d^{3}\right )} x^{4} - 10 \, {\left (4 \, a b^{6} c^{3} - 15 \, a^{2} b^{5} c^{2} d + 18 \, a^{3} b^{4} c d^{2} - 7 \, a^{4} b^{3} d^{3}\right )} x^{3} + 30 \, {\left (4 \, a^{2} b^{5} c^{3} - 15 \, a^{3} b^{4} c^{2} d + 18 \, a^{4} b^{3} c d^{2} - 7 \, a^{5} b^{2} d^{3}\right )} x^{2} + 180 \, {\left (a^{3} b^{4} c^{3} - 4 \, a^{4} b^{3} c^{2} d + 5 \, a^{5} b^{2} c d^{2} - 2 \, a^{6} b d^{3}\right )} x - 60 \, {\left (4 \, a^{4} b^{3} c^{3} - 15 \, a^{5} b^{2} c^{2} d + 18 \, a^{6} b c d^{2} - 7 \, a^{7} d^{3} + {\left (4 \, a^{3} b^{4} c^{3} - 15 \, a^{4} b^{3} c^{2} d + 18 \, a^{5} b^{2} c d^{2} - 7 \, a^{6} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{9} x + a b^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^3/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/60*(10*b^7*d^3*x^7 - 60*a^4*b^3*c^3 + 180*a^5*b^2*c^2*d - 180*a^6*b*c*d^2 + 60*a^7*d^3 + 2*(18*b^7*c*d^2 - 7
*a*b^6*d^3)*x^6 + 3*(15*b^7*c^2*d - 18*a*b^6*c*d^2 + 7*a^2*b^5*d^3)*x^5 + 5*(4*b^7*c^3 - 15*a*b^6*c^2*d + 18*a
^2*b^5*c*d^2 - 7*a^3*b^4*d^3)*x^4 - 10*(4*a*b^6*c^3 - 15*a^2*b^5*c^2*d + 18*a^3*b^4*c*d^2 - 7*a^4*b^3*d^3)*x^3
+ 30*(4*a^2*b^5*c^3 - 15*a^3*b^4*c^2*d + 18*a^4*b^3*c*d^2 - 7*a^5*b^2*d^3)*x^2 + 180*(a^3*b^4*c^3 - 4*a^4*b^3
*c^2*d + 5*a^5*b^2*c*d^2 - 2*a^6*b*d^3)*x - 60*(4*a^4*b^3*c^3 - 15*a^5*b^2*c^2*d + 18*a^6*b*c*d^2 - 7*a^7*d^3
+ (4*a^3*b^4*c^3 - 15*a^4*b^3*c^2*d + 18*a^5*b^2*c*d^2 - 7*a^6*b*d^3)*x)*log(b*x + a))/(b^9*x + a*b^8)

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giac [B]  time = 1.02, size = 403, normalized size = 2.02 \[ \frac {{\left (10 \, d^{3} + \frac {12 \, {\left (3 \, b^{2} c d^{2} - 7 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {45 \, {\left (b^{4} c^{2} d - 6 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {20 \, {\left (b^{6} c^{3} - 15 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 35 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} - \frac {30 \, {\left (4 \, a b^{7} c^{3} - 30 \, a^{2} b^{6} c^{2} d + 60 \, a^{3} b^{5} c d^{2} - 35 \, a^{4} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {180 \, {\left (2 \, a^{2} b^{8} c^{3} - 10 \, a^{3} b^{7} c^{2} d + 15 \, a^{4} b^{6} c d^{2} - 7 \, a^{5} b^{5} d^{3}\right )}}{{\left (b x + a\right )}^{5} b^{5}}\right )} {\left (b x + a\right )}^{6}}{60 \, b^{8}} + \frac {{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{8}} - \frac {\frac {a^{4} b^{9} c^{3}}{b x + a} - \frac {3 \, a^{5} b^{8} c^{2} d}{b x + a} + \frac {3 \, a^{6} b^{7} c d^{2}}{b x + a} - \frac {a^{7} b^{6} d^{3}}{b x + a}}{b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^3/(b*x+a)^2,x, algorithm="giac")

[Out]

1/60*(10*d^3 + 12*(3*b^2*c*d^2 - 7*a*b*d^3)/((b*x + a)*b) + 45*(b^4*c^2*d - 6*a*b^3*c*d^2 + 7*a^2*b^2*d^3)/((b
*x + a)^2*b^2) + 20*(b^6*c^3 - 15*a*b^5*c^2*d + 45*a^2*b^4*c*d^2 - 35*a^3*b^3*d^3)/((b*x + a)^3*b^3) - 30*(4*a
*b^7*c^3 - 30*a^2*b^6*c^2*d + 60*a^3*b^5*c*d^2 - 35*a^4*b^4*d^3)/((b*x + a)^4*b^4) + 180*(2*a^2*b^8*c^3 - 10*a
^3*b^7*c^2*d + 15*a^4*b^6*c*d^2 - 7*a^5*b^5*d^3)/((b*x + a)^5*b^5))*(b*x + a)^6/b^8 + (4*a^3*b^3*c^3 - 15*a^4*
b^2*c^2*d + 18*a^5*b*c*d^2 - 7*a^6*d^3)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^8 - (a^4*b^9*c^3/(b*x + a) -
3*a^5*b^8*c^2*d/(b*x + a) + 3*a^6*b^7*c*d^2/(b*x + a) - a^7*b^6*d^3/(b*x + a))/b^14

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maple [A]  time = 0.01, size = 378, normalized size = 1.89 \[ \frac {d^{3} x^{6}}{6 b^{2}}-\frac {2 a \,d^{3} x^{5}}{5 b^{3}}+\frac {3 c \,d^{2} x^{5}}{5 b^{2}}+\frac {3 a^{2} d^{3} x^{4}}{4 b^{4}}-\frac {3 a c \,d^{2} x^{4}}{2 b^{3}}+\frac {3 c^{2} d \,x^{4}}{4 b^{2}}-\frac {4 a^{3} d^{3} x^{3}}{3 b^{5}}+\frac {3 a^{2} c \,d^{2} x^{3}}{b^{4}}-\frac {2 a \,c^{2} d \,x^{3}}{b^{3}}+\frac {c^{3} x^{3}}{3 b^{2}}+\frac {5 a^{4} d^{3} x^{2}}{2 b^{6}}-\frac {6 a^{3} c \,d^{2} x^{2}}{b^{5}}+\frac {9 a^{2} c^{2} d \,x^{2}}{2 b^{4}}-\frac {a \,c^{3} x^{2}}{b^{3}}+\frac {a^{7} d^{3}}{\left (b x +a \right ) b^{8}}-\frac {3 a^{6} c \,d^{2}}{\left (b x +a \right ) b^{7}}+\frac {7 a^{6} d^{3} \ln \left (b x +a \right )}{b^{8}}+\frac {3 a^{5} c^{2} d}{\left (b x +a \right ) b^{6}}-\frac {18 a^{5} c \,d^{2} \ln \left (b x +a \right )}{b^{7}}-\frac {6 a^{5} d^{3} x}{b^{7}}-\frac {a^{4} c^{3}}{\left (b x +a \right ) b^{5}}+\frac {15 a^{4} c^{2} d \ln \left (b x +a \right )}{b^{6}}+\frac {15 a^{4} c \,d^{2} x}{b^{6}}-\frac {4 a^{3} c^{3} \ln \left (b x +a \right )}{b^{5}}-\frac {12 a^{3} c^{2} d x}{b^{5}}+\frac {3 a^{2} c^{3} x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d*x+c)^3/(b*x+a)^2,x)

[Out]

a^7/b^8/(b*x+a)*d^3-a^4/b^5/(b*x+a)*c^3-2/5/b^3*x^5*a*d^3+3/5/b^2*x^5*c*d^2+3/4/b^4*x^4*a^2*d^3+3/4/b^2*x^4*c^
2*d-4/3/b^5*x^3*a^3*d^3+5/2/b^6*x^2*a^4*d^3-1/b^3*x^2*a*c^3-6/b^7*x*a^5*d^3+3/b^4*x*a^2*c^3+7*a^6/b^8*ln(b*x+a
)*d^3-4*a^3/b^5*ln(b*x+a)*c^3-6/b^5*x^2*a^3*c*d^2+9/2/b^4*x^2*a^2*c^2*d+15/b^6*x*a^4*c*d^2+3/b^4*x^3*a^2*c*d^2
-3/2/b^3*x^4*a*c*d^2+15*a^4/b^6*ln(b*x+a)*c^2*d-3*a^6/b^7/(b*x+a)*c*d^2+3*a^5/b^6/(b*x+a)*c^2*d-2/b^3*x^3*a*c^
2*d-12/b^5*x*a^3*c^2*d-18*a^5/b^7*ln(b*x+a)*c*d^2+1/6*d^3*x^6/b^2+1/3/b^2*x^3*c^3

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maxima [A]  time = 1.09, size = 323, normalized size = 1.62 \[ -\frac {a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}}{b^{9} x + a b^{8}} + \frac {10 \, b^{5} d^{3} x^{6} + 12 \, {\left (3 \, b^{5} c d^{2} - 2 \, a b^{4} d^{3}\right )} x^{5} + 45 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{4} + 20 \, {\left (b^{5} c^{3} - 6 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 4 \, a^{3} b^{2} d^{3}\right )} x^{3} - 30 \, {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2} + 180 \, {\left (a^{2} b^{3} c^{3} - 4 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} x}{60 \, b^{7}} - \frac {{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (b x + a\right )}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x+c)^3/(b*x+a)^2,x, algorithm="maxima")

[Out]

-(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)/(b^9*x + a*b^8) + 1/60*(10*b^5*d^3*x^6 + 12*(3*b^5*
c*d^2 - 2*a*b^4*d^3)*x^5 + 45*(b^5*c^2*d - 2*a*b^4*c*d^2 + a^2*b^3*d^3)*x^4 + 20*(b^5*c^3 - 6*a*b^4*c^2*d + 9*
a^2*b^3*c*d^2 - 4*a^3*b^2*d^3)*x^3 - 30*(2*a*b^4*c^3 - 9*a^2*b^3*c^2*d + 12*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^2 +
180*(a^2*b^3*c^3 - 4*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 - 2*a^5*d^3)*x)/b^7 - (4*a^3*b^3*c^3 - 15*a^4*b^2*c^2*d +
18*a^5*b*c*d^2 - 7*a^6*d^3)*log(b*x + a)/b^8

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mupad [B]  time = 0.11, size = 688, normalized size = 3.44 \[ x^3\,\left (\frac {c^3}{3\,b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{3\,b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{3\,b^2}\right )-x^5\,\left (\frac {2\,a\,d^3}{5\,b^3}-\frac {3\,c\,d^2}{5\,b^2}\right )-x\,\left (\frac {a^2\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b^2}-\frac {2\,a\,\left (\frac {a^2\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b^2}+\frac {2\,a\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b}\right )}{b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{2\,b}-\frac {a^2\,d^3}{4\,b^4}\right )-x^2\,\left (\frac {a^2\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{2\,b^2}+\frac {a\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (7\,a^6\,d^3-18\,a^5\,b\,c\,d^2+15\,a^4\,b^2\,c^2\,d-4\,a^3\,b^3\,c^3\right )}{b^8}+\frac {a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d-a^4\,b^3\,c^3}{b\,\left (x\,b^8+a\,b^7\right )}+\frac {d^3\,x^6}{6\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(c + d*x)^3)/(a + b*x)^2,x)

[Out]

x^3*(c^3/(3*b^2) - (2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))/(3*b) + (a^
2*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/(3*b^2)) - x^5*((2*a*d^3)/(5*b^3) - (3*c*d^2)/(5*b^2)) - x*((a^2*(c^3/b^2 -
(2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))/b + (a^2*((2*a*d^3)/b^3 - (3*
c*d^2)/b^2))/b^2))/b^2 - (2*a*((a^2*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))
/b^2 + (2*a*(c^3/b^2 - (2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))/b + (a^
2*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b^2))/b))/b) + x^4*((3*c^2*d)/(4*b^2) + (a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))
/(2*b) - (a^2*d^3)/(4*b^4)) - x^2*((a^2*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b
^4))/(2*b^2) + (a*(c^3/b^2 - (2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))/b
+ (a^2*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b^2))/b) + (log(a + b*x)*(7*a^6*d^3 - 4*a^3*b^3*c^3 + 15*a^4*b^2*c^2*
d - 18*a^5*b*c*d^2))/b^8 + (a^7*d^3 - a^4*b^3*c^3 + 3*a^5*b^2*c^2*d - 3*a^6*b*c*d^2)/(b*(a*b^7 + b^8*x)) + (d^
3*x^6)/(6*b^2)

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sympy [A]  time = 1.22, size = 323, normalized size = 1.62 \[ \frac {a^{3} \left (a d - b c\right )^{2} \left (7 a d - 4 b c\right ) \log {\left (a + b x \right )}}{b^{8}} + x^{5} \left (- \frac {2 a d^{3}}{5 b^{3}} + \frac {3 c d^{2}}{5 b^{2}}\right ) + x^{4} \left (\frac {3 a^{2} d^{3}}{4 b^{4}} - \frac {3 a c d^{2}}{2 b^{3}} + \frac {3 c^{2} d}{4 b^{2}}\right ) + x^{3} \left (- \frac {4 a^{3} d^{3}}{3 b^{5}} + \frac {3 a^{2} c d^{2}}{b^{4}} - \frac {2 a c^{2} d}{b^{3}} + \frac {c^{3}}{3 b^{2}}\right ) + x^{2} \left (\frac {5 a^{4} d^{3}}{2 b^{6}} - \frac {6 a^{3} c d^{2}}{b^{5}} + \frac {9 a^{2} c^{2} d}{2 b^{4}} - \frac {a c^{3}}{b^{3}}\right ) + x \left (- \frac {6 a^{5} d^{3}}{b^{7}} + \frac {15 a^{4} c d^{2}}{b^{6}} - \frac {12 a^{3} c^{2} d}{b^{5}} + \frac {3 a^{2} c^{3}}{b^{4}}\right ) + \frac {a^{7} d^{3} - 3 a^{6} b c d^{2} + 3 a^{5} b^{2} c^{2} d - a^{4} b^{3} c^{3}}{a b^{8} + b^{9} x} + \frac {d^{3} x^{6}}{6 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(d*x+c)**3/(b*x+a)**2,x)

[Out]

a**3*(a*d - b*c)**2*(7*a*d - 4*b*c)*log(a + b*x)/b**8 + x**5*(-2*a*d**3/(5*b**3) + 3*c*d**2/(5*b**2)) + x**4*(
3*a**2*d**3/(4*b**4) - 3*a*c*d**2/(2*b**3) + 3*c**2*d/(4*b**2)) + x**3*(-4*a**3*d**3/(3*b**5) + 3*a**2*c*d**2/
b**4 - 2*a*c**2*d/b**3 + c**3/(3*b**2)) + x**2*(5*a**4*d**3/(2*b**6) - 6*a**3*c*d**2/b**5 + 9*a**2*c**2*d/(2*b
**4) - a*c**3/b**3) + x*(-6*a**5*d**3/b**7 + 15*a**4*c*d**2/b**6 - 12*a**3*c**2*d/b**5 + 3*a**2*c**3/b**4) + (
a**7*d**3 - 3*a**6*b*c*d**2 + 3*a**5*b**2*c**2*d - a**4*b**3*c**3)/(a*b**8 + b**9*x) + d**3*x**6/(6*b**2)

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