3.12.78 \(\int \frac {(c+d x)^7}{(a+b x)^2} \, dx\)

Optimal. Leaf size=187 \[ \frac {7 d^6 (a+b x)^5 (b c-a d)}{5 b^8}+\frac {21 d^5 (a+b x)^4 (b c-a d)^2}{4 b^8}+\frac {35 d^4 (a+b x)^3 (b c-a d)^3}{3 b^8}+\frac {35 d^3 (a+b x)^2 (b c-a d)^4}{2 b^8}-\frac {(b c-a d)^7}{b^8 (a+b x)}+\frac {7 d (b c-a d)^6 \log (a+b x)}{b^8}+\frac {d^7 (a+b x)^6}{6 b^8}+\frac {21 d^2 x (b c-a d)^5}{b^7} \]

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Rubi [A]  time = 0.23, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} \frac {7 d^6 (a+b x)^5 (b c-a d)}{5 b^8}+\frac {21 d^5 (a+b x)^4 (b c-a d)^2}{4 b^8}+\frac {35 d^4 (a+b x)^3 (b c-a d)^3}{3 b^8}+\frac {35 d^3 (a+b x)^2 (b c-a d)^4}{2 b^8}+\frac {21 d^2 x (b c-a d)^5}{b^7}-\frac {(b c-a d)^7}{b^8 (a+b x)}+\frac {7 d (b c-a d)^6 \log (a+b x)}{b^8}+\frac {d^7 (a+b x)^6}{6 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^2,x]

[Out]

(21*d^2*(b*c - a*d)^5*x)/b^7 - (b*c - a*d)^7/(b^8*(a + b*x)) + (35*d^3*(b*c - a*d)^4*(a + b*x)^2)/(2*b^8) + (3
5*d^4*(b*c - a*d)^3*(a + b*x)^3)/(3*b^8) + (21*d^5*(b*c - a*d)^2*(a + b*x)^4)/(4*b^8) + (7*d^6*(b*c - a*d)*(a
+ b*x)^5)/(5*b^8) + (d^7*(a + b*x)^6)/(6*b^8) + (7*d*(b*c - a*d)^6*Log[a + b*x])/b^8

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 {(c+d x)^7}{(a+b x)^2} \, dx &=\int \left (\frac {21 d^2 (b c-a d)^5}{b^7}+\frac {(b c-a d)^7}{b^7 (a+b x)^2}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)}+\frac {35 d^3 (b c-a d)^4 (a+b x)}{b^7}+\frac {35 d^4 (b c-a d)^3 (a+b x)^2}{b^7}+\frac {21 d^5 (b c-a d)^2 (a+b x)^3}{b^7}+\frac {7 d^6 (b c-a d) (a+b x)^4}{b^7}+\frac {d^7 (a+b x)^5}{b^7}\right ) \, dx\\ &=\frac {21 d^2 (b c-a d)^5 x}{b^7}-\frac {(b c-a d)^7}{b^8 (a+b x)}+\frac {35 d^3 (b c-a d)^4 (a+b x)^2}{2 b^8}+\frac {35 d^4 (b c-a d)^3 (a+b x)^3}{3 b^8}+\frac {21 d^5 (b c-a d)^2 (a+b x)^4}{4 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^5}{5 b^8}+\frac {d^7 (a+b x)^6}{6 b^8}+\frac {7 d (b c-a d)^6 \log (a+b x)}{b^8}\\ \end {align*}

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Mathematica [B]  time = 0.12, size = 388, normalized size = 2.07 \begin {gather*} \frac {60 a^7 d^7-60 a^6 b d^6 (7 c+6 d x)+210 a^5 b^2 d^5 \left (6 c^2+10 c d x-d^2 x^2\right )+70 a^4 b^3 d^4 \left (-30 c^3-72 c^2 d x+18 c d^2 x^2+d^3 x^3\right )-35 a^3 b^4 d^3 \left (-60 c^4-180 c^3 d x+90 c^2 d^2 x^2+12 c d^3 x^3+d^4 x^4\right )+21 a^2 b^5 d^2 \left (-60 c^5-200 c^4 d x+200 c^3 d^2 x^2+50 c^2 d^3 x^3+10 c d^4 x^4+d^5 x^5\right )-7 a b^6 d \left (-60 c^6-180 c^5 d x+450 c^4 d^2 x^2+200 c^3 d^3 x^3+75 c^2 d^4 x^4+18 c d^5 x^5+2 d^6 x^6\right )+420 d (a+b x) (b c-a d)^6 \log (a+b x)+b^7 \left (-60 c^7+1260 c^5 d^2 x^2+1050 c^4 d^3 x^3+700 c^3 d^4 x^4+315 c^2 d^5 x^5+84 c d^6 x^6+10 d^7 x^7\right )}{60 b^8 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^2,x]

[Out]

(60*a^7*d^7 - 60*a^6*b*d^6*(7*c + 6*d*x) + 210*a^5*b^2*d^5*(6*c^2 + 10*c*d*x - d^2*x^2) + 70*a^4*b^3*d^4*(-30*
c^3 - 72*c^2*d*x + 18*c*d^2*x^2 + d^3*x^3) - 35*a^3*b^4*d^3*(-60*c^4 - 180*c^3*d*x + 90*c^2*d^2*x^2 + 12*c*d^3
*x^3 + d^4*x^4) + 21*a^2*b^5*d^2*(-60*c^5 - 200*c^4*d*x + 200*c^3*d^2*x^2 + 50*c^2*d^3*x^3 + 10*c*d^4*x^4 + d^
5*x^5) - 7*a*b^6*d*(-60*c^6 - 180*c^5*d*x + 450*c^4*d^2*x^2 + 200*c^3*d^3*x^3 + 75*c^2*d^4*x^4 + 18*c*d^5*x^5
+ 2*d^6*x^6) + b^7*(-60*c^7 + 1260*c^5*d^2*x^2 + 1050*c^4*d^3*x^3 + 700*c^3*d^4*x^4 + 315*c^2*d^5*x^5 + 84*c*d
^6*x^6 + 10*d^7*x^7) + 420*d*(b*c - a*d)^6*(a + b*x)*Log[a + b*x])/(60*b^8*(a + b*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^7}{(a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^2,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^2, x]

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fricas [B]  time = 1.58, size = 632, normalized size = 3.38 \begin {gather*} \frac {10 \, b^{7} d^{7} x^{7} - 60 \, b^{7} c^{7} + 420 \, a b^{6} c^{6} d - 1260 \, a^{2} b^{5} c^{5} d^{2} + 2100 \, a^{3} b^{4} c^{4} d^{3} - 2100 \, a^{4} b^{3} c^{3} d^{4} + 1260 \, a^{5} b^{2} c^{2} d^{5} - 420 \, a^{6} b c d^{6} + 60 \, a^{7} d^{7} + 14 \, {\left (6 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 21 \, {\left (15 \, b^{7} c^{2} d^{5} - 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 35 \, {\left (20 \, b^{7} c^{3} d^{4} - 15 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 70 \, {\left (15 \, b^{7} c^{4} d^{3} - 20 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} - 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 210 \, {\left (6 \, b^{7} c^{5} d^{2} - 15 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 15 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 60 \, {\left (21 \, a b^{6} c^{5} d^{2} - 70 \, a^{2} b^{5} c^{4} d^{3} + 105 \, a^{3} b^{4} c^{3} d^{4} - 84 \, a^{4} b^{3} c^{2} d^{5} + 35 \, a^{5} b^{2} c d^{6} - 6 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a b^{6} c^{6} d - 6 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} - 20 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} - 6 \, a^{6} b c d^{6} + a^{7} d^{7} + {\left (b^{7} c^{6} d - 6 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} - 20 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} - 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{9} x + a b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*b^7*d^7*x^7 - 60*b^7*c^7 + 420*a*b^6*c^6*d - 1260*a^2*b^5*c^5*d^2 + 2100*a^3*b^4*c^4*d^3 - 2100*a^4*b
^3*c^3*d^4 + 1260*a^5*b^2*c^2*d^5 - 420*a^6*b*c*d^6 + 60*a^7*d^7 + 14*(6*b^7*c*d^6 - a*b^6*d^7)*x^6 + 21*(15*b
^7*c^2*d^5 - 6*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 35*(20*b^7*c^3*d^4 - 15*a*b^6*c^2*d^5 + 6*a^2*b^5*c*d^6 - a^3*
b^4*d^7)*x^4 + 70*(15*b^7*c^4*d^3 - 20*a*b^6*c^3*d^4 + 15*a^2*b^5*c^2*d^5 - 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3
+ 210*(6*b^7*c^5*d^2 - 15*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 - 15*a^3*b^4*c^2*d^5 + 6*a^4*b^3*c*d^6 - a^5*b^2
*d^7)*x^2 + 60*(21*a*b^6*c^5*d^2 - 70*a^2*b^5*c^4*d^3 + 105*a^3*b^4*c^3*d^4 - 84*a^4*b^3*c^2*d^5 + 35*a^5*b^2*
c*d^6 - 6*a^6*b*d^7)*x + 420*(a*b^6*c^6*d - 6*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 - 20*a^4*b^3*c^3*d^4 + 15*a
^5*b^2*c^2*d^5 - 6*a^6*b*c*d^6 + a^7*d^7 + (b^7*c^6*d - 6*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 - 20*a^3*b^4*c^3*
d^4 + 15*a^4*b^3*c^2*d^5 - 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a))/(b^9*x + a*b^8)

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giac [B]  time = 1.28, size = 567, normalized size = 3.03 \begin {gather*} \frac {{\left (10 \, d^{7} + \frac {84 \, {\left (b^{2} c d^{6} - a b d^{7}\right )}}{{\left (b x + a\right )} b} + \frac {315 \, {\left (b^{4} c^{2} d^{5} - 2 \, a b^{3} c d^{6} + a^{2} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {700 \, {\left (b^{6} c^{3} d^{4} - 3 \, a b^{5} c^{2} d^{5} + 3 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {1050 \, {\left (b^{8} c^{4} d^{3} - 4 \, a b^{7} c^{3} d^{4} + 6 \, a^{2} b^{6} c^{2} d^{5} - 4 \, a^{3} b^{5} c d^{6} + a^{4} b^{4} d^{7}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {1260 \, {\left (b^{10} c^{5} d^{2} - 5 \, a b^{9} c^{4} d^{3} + 10 \, a^{2} b^{8} c^{3} d^{4} - 10 \, a^{3} b^{7} c^{2} d^{5} + 5 \, a^{4} b^{6} c d^{6} - a^{5} b^{5} d^{7}\right )}}{{\left (b x + a\right )}^{5} b^{5}}\right )} {\left (b x + a\right )}^{6}}{60 \, b^{8}} - \frac {7 \, {\left (b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} - 20 \, a^{3} b^{3} c^{3} d^{4} + 15 \, a^{4} b^{2} c^{2} d^{5} - 6 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{8}} - \frac {\frac {b^{13} c^{7}}{b x + a} - \frac {7 \, a b^{12} c^{6} d}{b x + a} + \frac {21 \, a^{2} b^{11} c^{5} d^{2}}{b x + a} - \frac {35 \, a^{3} b^{10} c^{4} d^{3}}{b x + a} + \frac {35 \, a^{4} b^{9} c^{3} d^{4}}{b x + a} - \frac {21 \, a^{5} b^{8} c^{2} d^{5}}{b x + a} + \frac {7 \, a^{6} b^{7} c d^{6}}{b x + a} - \frac {a^{7} b^{6} d^{7}}{b x + a}}{b^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*d^7 + 84*(b^2*c*d^6 - a*b*d^7)/((b*x + a)*b) + 315*(b^4*c^2*d^5 - 2*a*b^3*c*d^6 + a^2*b^2*d^7)/((b*x
+ a)^2*b^2) + 700*(b^6*c^3*d^4 - 3*a*b^5*c^2*d^5 + 3*a^2*b^4*c*d^6 - a^3*b^3*d^7)/((b*x + a)^3*b^3) + 1050*(b^
8*c^4*d^3 - 4*a*b^7*c^3*d^4 + 6*a^2*b^6*c^2*d^5 - 4*a^3*b^5*c*d^6 + a^4*b^4*d^7)/((b*x + a)^4*b^4) + 1260*(b^1
0*c^5*d^2 - 5*a*b^9*c^4*d^3 + 10*a^2*b^8*c^3*d^4 - 10*a^3*b^7*c^2*d^5 + 5*a^4*b^6*c*d^6 - a^5*b^5*d^7)/((b*x +
a)^5*b^5))*(b*x + a)^6/b^8 - 7*(b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^
4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^8 - (b^13*c^7/(b*x + a) - 7*
a*b^12*c^6*d/(b*x + a) + 21*a^2*b^11*c^5*d^2/(b*x + a) - 35*a^3*b^10*c^4*d^3/(b*x + a) + 35*a^4*b^9*c^3*d^4/(b
*x + a) - 21*a^5*b^8*c^2*d^5/(b*x + a) + 7*a^6*b^7*c*d^6/(b*x + a) - a^7*b^6*d^7/(b*x + a))/b^14

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maple [B]  time = 0.01, size = 571, normalized size = 3.05 \begin {gather*} \frac {d^{7} x^{6}}{6 b^{2}}-\frac {2 a \,d^{7} x^{5}}{5 b^{3}}+\frac {7 c \,d^{6} x^{5}}{5 b^{2}}+\frac {3 a^{2} d^{7} x^{4}}{4 b^{4}}-\frac {7 a c \,d^{6} x^{4}}{2 b^{3}}+\frac {21 c^{2} d^{5} x^{4}}{4 b^{2}}-\frac {4 a^{3} d^{7} x^{3}}{3 b^{5}}+\frac {7 a^{2} c \,d^{6} x^{3}}{b^{4}}-\frac {14 a \,c^{2} d^{5} x^{3}}{b^{3}}+\frac {35 c^{3} d^{4} x^{3}}{3 b^{2}}+\frac {5 a^{4} d^{7} x^{2}}{2 b^{6}}-\frac {14 a^{3} c \,d^{6} x^{2}}{b^{5}}+\frac {63 a^{2} c^{2} d^{5} x^{2}}{2 b^{4}}-\frac {35 a \,c^{3} d^{4} x^{2}}{b^{3}}+\frac {35 c^{4} d^{3} x^{2}}{2 b^{2}}+\frac {a^{7} d^{7}}{\left (b x +a \right ) b^{8}}-\frac {7 a^{6} c \,d^{6}}{\left (b x +a \right ) b^{7}}+\frac {7 a^{6} d^{7} \ln \left (b x +a \right )}{b^{8}}+\frac {21 a^{5} c^{2} d^{5}}{\left (b x +a \right ) b^{6}}-\frac {42 a^{5} c \,d^{6} \ln \left (b x +a \right )}{b^{7}}-\frac {6 a^{5} d^{7} x}{b^{7}}-\frac {35 a^{4} c^{3} d^{4}}{\left (b x +a \right ) b^{5}}+\frac {105 a^{4} c^{2} d^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {35 a^{4} c \,d^{6} x}{b^{6}}+\frac {35 a^{3} c^{4} d^{3}}{\left (b x +a \right ) b^{4}}-\frac {140 a^{3} c^{3} d^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {84 a^{3} c^{2} d^{5} x}{b^{5}}-\frac {21 a^{2} c^{5} d^{2}}{\left (b x +a \right ) b^{3}}+\frac {105 a^{2} c^{4} d^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {105 a^{2} c^{3} d^{4} x}{b^{4}}+\frac {7 a \,c^{6} d}{\left (b x +a \right ) b^{2}}-\frac {42 a \,c^{5} d^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {70 a \,c^{4} d^{3} x}{b^{3}}-\frac {c^{7}}{\left (b x +a \right ) b}+\frac {7 c^{6} d \ln \left (b x +a \right )}{b^{2}}+\frac {21 c^{5} d^{2} x}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/3*d^4/b^2*x^3*c^3+5/2*d^7/b^6*x^2*a^4+35/2*d^3/b^2*x^2*c^4-6*d^7/b^7*a^5*x+21*d^2/b^2*c^5*x+7/b^8*d^7*ln(b*
x+a)*a^6+7/b^2*d*ln(b*x+a)*c^6+1/b^8/(b*x+a)*a^7*d^7-2/5*d^7/b^3*x^5*a+7/5*d^6/b^2*x^5*c+3/4*d^7/b^4*x^4*a^2+2
1/4*d^5/b^2*x^4*c^2-4/3*d^7/b^5*x^3*a^3-21/b^3/(b*x+a)*a^2*c^5*d^2+7/b^2/(b*x+a)*a*c^6*d-14*d^5/b^3*x^3*a*c^2-
14*d^6/b^5*x^2*a^3*c+63/2*d^5/b^4*x^2*a^2*c^2-35*d^4/b^3*x^2*a*c^3+35*d^6/b^6*a^4*c*x-7/2*d^6/b^3*x^4*a*c+7*d^
6/b^4*x^3*a^2*c-70*d^3/b^3*a*c^4*x-42/b^7*d^6*ln(b*x+a)*a^5*c+105/b^6*d^5*ln(b*x+a)*a^4*c^2-140/b^5*d^4*ln(b*x
+a)*a^3*c^3+105/b^4*d^3*ln(b*x+a)*a^2*c^4-42/b^3*d^2*ln(b*x+a)*a*c^5-7/b^7/(b*x+a)*a^6*c*d^6+21/b^6/(b*x+a)*a^
5*c^2*d^5-35/b^5/(b*x+a)*a^4*c^3*d^4+35/b^4/(b*x+a)*a^3*c^4*d^3-84*d^5/b^5*a^3*c^2*x+105*d^4/b^4*a^2*c^3*x-1/b
/(b*x+a)*c^7+1/6*d^7/b^2*x^6

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maxima [B]  time = 1.40, size = 467, normalized size = 2.50 \begin {gather*} -\frac {b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}}{b^{9} x + a b^{8}} + \frac {10 \, b^{5} d^{7} x^{6} + 12 \, {\left (7 \, b^{5} c d^{6} - 2 \, a b^{4} d^{7}\right )} x^{5} + 15 \, {\left (21 \, b^{5} c^{2} d^{5} - 14 \, a b^{4} c d^{6} + 3 \, a^{2} b^{3} d^{7}\right )} x^{4} + 20 \, {\left (35 \, b^{5} c^{3} d^{4} - 42 \, a b^{4} c^{2} d^{5} + 21 \, a^{2} b^{3} c d^{6} - 4 \, a^{3} b^{2} d^{7}\right )} x^{3} + 30 \, {\left (35 \, b^{5} c^{4} d^{3} - 70 \, a b^{4} c^{3} d^{4} + 63 \, a^{2} b^{3} c^{2} d^{5} - 28 \, a^{3} b^{2} c d^{6} + 5 \, a^{4} b d^{7}\right )} x^{2} + 60 \, {\left (21 \, b^{5} c^{5} d^{2} - 70 \, a b^{4} c^{4} d^{3} + 105 \, a^{2} b^{3} c^{3} d^{4} - 84 \, a^{3} b^{2} c^{2} d^{5} + 35 \, a^{4} b c d^{6} - 6 \, a^{5} d^{7}\right )} x}{60 \, b^{7}} + \frac {7 \, {\left (b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 15 \, a^{2} b^{4} c^{4} d^{3} - 20 \, a^{3} b^{3} c^{3} d^{4} + 15 \, a^{4} b^{2} c^{2} d^{5} - 6 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5
+ 7*a^6*b*c*d^6 - a^7*d^7)/(b^9*x + a*b^8) + 1/60*(10*b^5*d^7*x^6 + 12*(7*b^5*c*d^6 - 2*a*b^4*d^7)*x^5 + 15*(2
1*b^5*c^2*d^5 - 14*a*b^4*c*d^6 + 3*a^2*b^3*d^7)*x^4 + 20*(35*b^5*c^3*d^4 - 42*a*b^4*c^2*d^5 + 21*a^2*b^3*c*d^6
- 4*a^3*b^2*d^7)*x^3 + 30*(35*b^5*c^4*d^3 - 70*a*b^4*c^3*d^4 + 63*a^2*b^3*c^2*d^5 - 28*a^3*b^2*c*d^6 + 5*a^4*
b*d^7)*x^2 + 60*(21*b^5*c^5*d^2 - 70*a*b^4*c^4*d^3 + 105*a^2*b^3*c^3*d^4 - 84*a^3*b^2*c^2*d^5 + 35*a^4*b*c*d^6
- 6*a^5*d^7)*x)/b^7 + 7*(b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c
^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)*log(b*x + a)/b^8

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mupad [B]  time = 0.24, size = 841, normalized size = 4.50 \begin {gather*} x^4\,\left (\frac {a\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{2\,b}-\frac {a^2\,d^7}{4\,b^4}+\frac {21\,c^2\,d^5}{4\,b^2}\right )-x^2\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b}-\frac {a^2\,d^7}{b^4}+\frac {21\,c^2\,d^5}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b^2}\right )}{b}-\frac {35\,c^4\,d^3}{2\,b^2}+\frac {a^2\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b}-\frac {a^2\,d^7}{b^4}+\frac {21\,c^2\,d^5}{b^2}\right )}{2\,b^2}\right )-x^5\,\left (\frac {2\,a\,d^7}{5\,b^3}-\frac {7\,c\,d^6}{5\,b^2}\right )+x\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {35\,c^3\,d^4}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b}-\frac {a^2\,d^7}{b^4}+\frac {21\,c^2\,d^5}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b^2}\right )}{b}-\frac {35\,c^4\,d^3}{b^2}+\frac {a^2\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b}-\frac {a^2\,d^7}{b^4}+\frac {21\,c^2\,d^5}{b^2}\right )}{b^2}\right )}{b}-\frac {a^2\,\left (\frac {35\,c^3\,d^4}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b}-\frac {a^2\,d^7}{b^4}+\frac {21\,c^2\,d^5}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b^2}\right )}{b^2}+\frac {21\,c^5\,d^2}{b^2}\right )+x^3\,\left (\frac {35\,c^3\,d^4}{3\,b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{b}-\frac {a^2\,d^7}{b^4}+\frac {21\,c^2\,d^5}{b^2}\right )}{3\,b}+\frac {a^2\,\left (\frac {2\,a\,d^7}{b^3}-\frac {7\,c\,d^6}{b^2}\right )}{3\,b^2}\right )+\frac {a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7}{b\,\left (x\,b^8+a\,b^7\right )}+\frac {d^7\,x^6}{6\,b^2}+\frac {\ln \left (a+b\,x\right )\,\left (7\,a^6\,d^7-42\,a^5\,b\,c\,d^6+105\,a^4\,b^2\,c^2\,d^5-140\,a^3\,b^3\,c^3\,d^4+105\,a^2\,b^4\,c^4\,d^3-42\,a\,b^5\,c^5\,d^2+7\,b^6\,c^6\,d\right )}{b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^2,x)

[Out]

x^4*((a*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/(2*b) - (a^2*d^7)/(4*b^4) + (21*c^2*d^5)/(4*b^2)) - x^2*((a*((35*c^3*
d^4)/b^2 - (2*a*((2*a*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/b - (a^2*d^7)/b^4 + (21*c^2*d^5)/b^2))/b + (a^2*((2*a*d
^7)/b^3 - (7*c*d^6)/b^2))/b^2))/b - (35*c^4*d^3)/(2*b^2) + (a^2*((2*a*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/b - (a^
2*d^7)/b^4 + (21*c^2*d^5)/b^2))/(2*b^2)) - x^5*((2*a*d^7)/(5*b^3) - (7*c*d^6)/(5*b^2)) + x*((2*a*((2*a*((35*c^
3*d^4)/b^2 - (2*a*((2*a*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/b - (a^2*d^7)/b^4 + (21*c^2*d^5)/b^2))/b + (a^2*((2*a
*d^7)/b^3 - (7*c*d^6)/b^2))/b^2))/b - (35*c^4*d^3)/b^2 + (a^2*((2*a*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/b - (a^2*
d^7)/b^4 + (21*c^2*d^5)/b^2))/b^2))/b - (a^2*((35*c^3*d^4)/b^2 - (2*a*((2*a*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/b
- (a^2*d^7)/b^4 + (21*c^2*d^5)/b^2))/b + (a^2*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/b^2))/b^2 + (21*c^5*d^2)/b^2)
+ x^3*((35*c^3*d^4)/(3*b^2) - (2*a*((2*a*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/b - (a^2*d^7)/b^4 + (21*c^2*d^5)/b^2
))/(3*b) + (a^2*((2*a*d^7)/b^3 - (7*c*d^6)/b^2))/(3*b^2)) + (a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b
^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)/(b*(a*b^7 + b^8*x)) + (d
^7*x^6)/(6*b^2) + (log(a + b*x)*(7*a^6*d^7 + 7*b^6*c^6*d - 42*a*b^5*c^5*d^2 + 105*a^2*b^4*c^4*d^3 - 140*a^3*b^
3*c^3*d^4 + 105*a^4*b^2*c^2*d^5 - 42*a^5*b*c*d^6))/b^8

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sympy [B]  time = 1.44, size = 428, normalized size = 2.29 \begin {gather*} x^{5} \left (- \frac {2 a d^{7}}{5 b^{3}} + \frac {7 c d^{6}}{5 b^{2}}\right ) + x^{4} \left (\frac {3 a^{2} d^{7}}{4 b^{4}} - \frac {7 a c d^{6}}{2 b^{3}} + \frac {21 c^{2} d^{5}}{4 b^{2}}\right ) + x^{3} \left (- \frac {4 a^{3} d^{7}}{3 b^{5}} + \frac {7 a^{2} c d^{6}}{b^{4}} - \frac {14 a c^{2} d^{5}}{b^{3}} + \frac {35 c^{3} d^{4}}{3 b^{2}}\right ) + x^{2} \left (\frac {5 a^{4} d^{7}}{2 b^{6}} - \frac {14 a^{3} c d^{6}}{b^{5}} + \frac {63 a^{2} c^{2} d^{5}}{2 b^{4}} - \frac {35 a c^{3} d^{4}}{b^{3}} + \frac {35 c^{4} d^{3}}{2 b^{2}}\right ) + x \left (- \frac {6 a^{5} d^{7}}{b^{7}} + \frac {35 a^{4} c d^{6}}{b^{6}} - \frac {84 a^{3} c^{2} d^{5}}{b^{5}} + \frac {105 a^{2} c^{3} d^{4}}{b^{4}} - \frac {70 a c^{4} d^{3}}{b^{3}} + \frac {21 c^{5} d^{2}}{b^{2}}\right ) + \frac {a^{7} d^{7} - 7 a^{6} b c d^{6} + 21 a^{5} b^{2} c^{2} d^{5} - 35 a^{4} b^{3} c^{3} d^{4} + 35 a^{3} b^{4} c^{4} d^{3} - 21 a^{2} b^{5} c^{5} d^{2} + 7 a b^{6} c^{6} d - b^{7} c^{7}}{a b^{8} + b^{9} x} + \frac {d^{7} x^{6}}{6 b^{2}} + \frac {7 d \left (a d - b c\right )^{6} \log {\left (a + b x \right )}}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5*(-2*a*d**7/(5*b**3) + 7*c*d**6/(5*b**2)) + x**4*(3*a**2*d**7/(4*b**4) - 7*a*c*d**6/(2*b**3) + 21*c**2*d**
5/(4*b**2)) + x**3*(-4*a**3*d**7/(3*b**5) + 7*a**2*c*d**6/b**4 - 14*a*c**2*d**5/b**3 + 35*c**3*d**4/(3*b**2))
+ x**2*(5*a**4*d**7/(2*b**6) - 14*a**3*c*d**6/b**5 + 63*a**2*c**2*d**5/(2*b**4) - 35*a*c**3*d**4/b**3 + 35*c**
4*d**3/(2*b**2)) + x*(-6*a**5*d**7/b**7 + 35*a**4*c*d**6/b**6 - 84*a**3*c**2*d**5/b**5 + 105*a**2*c**3*d**4/b*
*4 - 70*a*c**4*d**3/b**3 + 21*c**5*d**2/b**2) + (a**7*d**7 - 7*a**6*b*c*d**6 + 21*a**5*b**2*c**2*d**5 - 35*a**
4*b**3*c**3*d**4 + 35*a**3*b**4*c**4*d**3 - 21*a**2*b**5*c**5*d**2 + 7*a*b**6*c**6*d - b**7*c**7)/(a*b**8 + b*
*9*x) + d**7*x**6/(6*b**2) + 7*d*(a*d - b*c)**6*log(a + b*x)/b**8

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