3.12.82 \(\int \frac {(c+d x)^7}{(a+b x)^6} \, dx\)

Optimal. Leaf size=181 \[ \frac {21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}-\frac {35 d^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac {7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}-\frac {7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac {(b c-a d)^7}{5 b^8 (a+b x)^5}+\frac {d^6 x (7 b c-6 a d)}{b^7}+\frac {d^7 x^2}{2 b^6} \]

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Rubi [A]  time = 0.19, antiderivative size = 181, 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 {d^6 x (7 b c-6 a d)}{b^7}-\frac {35 d^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac {7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}+\frac {21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}-\frac {7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac {(b c-a d)^7}{5 b^8 (a+b x)^5}+\frac {d^7 x^2}{2 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^6,x]

[Out]

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

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Mathematica [B]  time = 0.15, size = 389, normalized size = 2.15 \begin {gather*} \frac {459 a^7 d^7+3 a^6 b d^6 (625 d x-406 c)+a^5 b^2 d^5 \left (959 c^2-5250 c d x+2700 d^2 x^2\right )+5 a^4 b^3 d^4 \left (-28 c^3+875 c^2 d x-1680 c d^2 x^2+260 d^3 x^3\right )-5 a^3 b^4 d^3 \left (7 c^4+140 c^3 d x-1540 c^2 d^2 x^2+1120 c d^3 x^3+80 d^4 x^4\right )-a^2 b^5 d^2 \left (14 c^5+175 c^4 d x+1400 c^3 d^2 x^2-6300 c^2 d^3 x^3+700 c d^4 x^4+500 d^5 x^5\right )-7 a b^6 d \left (c^6+10 c^5 d x+50 c^4 d^2 x^2+200 c^3 d^3 x^3-300 c^2 d^4 x^4-100 c d^5 x^5+10 d^6 x^6\right )+420 d^5 (a+b x)^5 (b c-a d)^2 \log (a+b x)-\left (b^7 \left (4 c^7+35 c^6 d x+140 c^5 d^2 x^2+350 c^4 d^3 x^3+700 c^3 d^4 x^4-140 c d^6 x^6-10 d^7 x^7\right )\right )}{20 b^8 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^6,x]

[Out]

(459*a^7*d^7 + 3*a^6*b*d^6*(-406*c + 625*d*x) + a^5*b^2*d^5*(959*c^2 - 5250*c*d*x + 2700*d^2*x^2) + 5*a^4*b^3*
d^4*(-28*c^3 + 875*c^2*d*x - 1680*c*d^2*x^2 + 260*d^3*x^3) - 5*a^3*b^4*d^3*(7*c^4 + 140*c^3*d*x - 1540*c^2*d^2
*x^2 + 1120*c*d^3*x^3 + 80*d^4*x^4) - a^2*b^5*d^2*(14*c^5 + 175*c^4*d*x + 1400*c^3*d^2*x^2 - 6300*c^2*d^3*x^3
+ 700*c*d^4*x^4 + 500*d^5*x^5) - 7*a*b^6*d*(c^6 + 10*c^5*d*x + 50*c^4*d^2*x^2 + 200*c^3*d^3*x^3 - 300*c^2*d^4*
x^4 - 100*c*d^5*x^5 + 10*d^6*x^6) - b^7*(4*c^7 + 35*c^6*d*x + 140*c^5*d^2*x^2 + 350*c^4*d^3*x^3 + 700*c^3*d^4*
x^4 - 140*c*d^6*x^6 - 10*d^7*x^7) + 420*d^5*(b*c - a*d)^2*(a + b*x)^5*Log[a + b*x])/(20*b^8*(a + b*x)^5)

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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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^6,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^6, x]

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fricas [B]  time = 1.40, size = 732, normalized size = 4.04 \begin {gather*} \frac {10 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 14 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 140 \, a^{4} b^{3} c^{3} d^{4} + 959 \, a^{5} b^{2} c^{2} d^{5} - 1218 \, a^{6} b c d^{6} + 459 \, a^{7} d^{7} + 70 \, {\left (2 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 100 \, {\left (7 \, a b^{6} c d^{6} - 5 \, a^{2} b^{5} d^{7}\right )} x^{5} - 100 \, {\left (7 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + 4 \, a^{3} b^{4} d^{7}\right )} x^{4} - 50 \, {\left (7 \, b^{7} c^{4} d^{3} + 28 \, a b^{6} c^{3} d^{4} - 126 \, a^{2} b^{5} c^{2} d^{5} + 112 \, a^{3} b^{4} c d^{6} - 26 \, a^{4} b^{3} d^{7}\right )} x^{3} - 10 \, {\left (14 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 140 \, a^{2} b^{5} c^{3} d^{4} - 770 \, a^{3} b^{4} c^{2} d^{5} + 840 \, a^{4} b^{3} c d^{6} - 270 \, a^{5} b^{2} d^{7}\right )} x^{2} - 5 \, {\left (7 \, b^{7} c^{6} d + 14 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 140 \, a^{3} b^{4} c^{3} d^{4} - 875 \, a^{4} b^{3} c^{2} d^{5} + 1050 \, a^{5} b^{2} c d^{6} - 375 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{5} b^{2} c^{2} d^{5} - 2 \, a^{6} b c d^{6} + a^{7} d^{7} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 5 \, {\left (a b^{6} c^{2} d^{5} - 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{5} c^{2} d^{5} - 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 10 \, {\left (a^{3} b^{4} c^{2} d^{5} - 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 5 \, {\left (a^{4} b^{3} c^{2} d^{5} - 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^6,x, algorithm="fricas")

[Out]

1/20*(10*b^7*d^7*x^7 - 4*b^7*c^7 - 7*a*b^6*c^6*d - 14*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 - 140*a^4*b^3*c^3*d
^4 + 959*a^5*b^2*c^2*d^5 - 1218*a^6*b*c*d^6 + 459*a^7*d^7 + 70*(2*b^7*c*d^6 - a*b^6*d^7)*x^6 + 100*(7*a*b^6*c*
d^6 - 5*a^2*b^5*d^7)*x^5 - 100*(7*b^7*c^3*d^4 - 21*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 + 4*a^3*b^4*d^7)*x^4 - 50*(
7*b^7*c^4*d^3 + 28*a*b^6*c^3*d^4 - 126*a^2*b^5*c^2*d^5 + 112*a^3*b^4*c*d^6 - 26*a^4*b^3*d^7)*x^3 - 10*(14*b^7*
c^5*d^2 + 35*a*b^6*c^4*d^3 + 140*a^2*b^5*c^3*d^4 - 770*a^3*b^4*c^2*d^5 + 840*a^4*b^3*c*d^6 - 270*a^5*b^2*d^7)*
x^2 - 5*(7*b^7*c^6*d + 14*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 140*a^3*b^4*c^3*d^4 - 875*a^4*b^3*c^2*d^5 + 105
0*a^5*b^2*c*d^6 - 375*a^6*b*d^7)*x + 420*(a^5*b^2*c^2*d^5 - 2*a^6*b*c*d^6 + a^7*d^7 + (b^7*c^2*d^5 - 2*a*b^6*c
*d^6 + a^2*b^5*d^7)*x^5 + 5*(a*b^6*c^2*d^5 - 2*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 10*(a^2*b^5*c^2*d^5 - 2*a^3*
b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 10*(a^3*b^4*c^2*d^5 - 2*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 5*(a^4*b^3*c^2*d^5 -
2*a^5*b^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 +
5*a^4*b^9*x + a^5*b^8)

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giac [B]  time = 1.38, size = 463, normalized size = 2.56 \begin {gather*} \frac {21 \, {\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac {b^{6} d^{7} x^{2} + 14 \, b^{6} c d^{6} x - 12 \, a b^{5} d^{7} x}{2 \, b^{12}} - \frac {4 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 14 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} - 959 \, a^{5} b^{2} c^{2} d^{5} + 1218 \, a^{6} b c d^{6} - 459 \, a^{7} d^{7} + 700 \, {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 350 \, {\left (b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} - 18 \, a^{2} b^{5} c^{2} d^{5} + 20 \, a^{3} b^{4} c d^{6} - 7 \, a^{4} b^{3} d^{7}\right )} x^{3} + 70 \, {\left (2 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 110 \, a^{3} b^{4} c^{2} d^{5} + 130 \, a^{4} b^{3} c d^{6} - 47 \, a^{5} b^{2} d^{7}\right )} x^{2} + 35 \, {\left (b^{7} c^{6} d + 2 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} - 125 \, a^{4} b^{3} c^{2} d^{5} + 154 \, a^{5} b^{2} c d^{6} - 57 \, a^{6} b d^{7}\right )} x}{20 \, {\left (b x + a\right )}^{5} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^6,x, algorithm="giac")

[Out]

21*(b^2*c^2*d^5 - 2*a*b*c*d^6 + a^2*d^7)*log(abs(b*x + a))/b^8 + 1/2*(b^6*d^7*x^2 + 14*b^6*c*d^6*x - 12*a*b^5*
d^7*x)/b^12 - 1/20*(4*b^7*c^7 + 7*a*b^6*c^6*d + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4
- 959*a^5*b^2*c^2*d^5 + 1218*a^6*b*c*d^6 - 459*a^7*d^7 + 700*(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6
- a^3*b^4*d^7)*x^4 + 350*(b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 - 18*a^2*b^5*c^2*d^5 + 20*a^3*b^4*c*d^6 - 7*a^4*b^3*d^
7)*x^3 + 70*(2*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 - 110*a^3*b^4*c^2*d^5 + 130*a^4*b^3*c*d^6 -
47*a^5*b^2*d^7)*x^2 + 35*(b^7*c^6*d + 2*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 - 125*a^4*b^3*c
^2*d^5 + 154*a^5*b^2*c*d^6 - 57*a^6*b*d^7)*x)/((b*x + a)^5*b^8)

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maple [B]  time = 0.01, size = 656, normalized size = 3.62 \begin {gather*} \frac {a^{7} d^{7}}{5 \left (b x +a \right )^{5} b^{8}}-\frac {7 a^{6} c \,d^{6}}{5 \left (b x +a \right )^{5} b^{7}}+\frac {21 a^{5} c^{2} d^{5}}{5 \left (b x +a \right )^{5} b^{6}}-\frac {7 a^{4} c^{3} d^{4}}{\left (b x +a \right )^{5} b^{5}}+\frac {7 a^{3} c^{4} d^{3}}{\left (b x +a \right )^{5} b^{4}}-\frac {21 a^{2} c^{5} d^{2}}{5 \left (b x +a \right )^{5} b^{3}}+\frac {7 a \,c^{6} d}{5 \left (b x +a \right )^{5} b^{2}}-\frac {c^{7}}{5 \left (b x +a \right )^{5} b}-\frac {7 a^{6} d^{7}}{4 \left (b x +a \right )^{4} b^{8}}+\frac {21 a^{5} c \,d^{6}}{2 \left (b x +a \right )^{4} b^{7}}-\frac {105 a^{4} c^{2} d^{5}}{4 \left (b x +a \right )^{4} b^{6}}+\frac {35 a^{3} c^{3} d^{4}}{\left (b x +a \right )^{4} b^{5}}-\frac {105 a^{2} c^{4} d^{3}}{4 \left (b x +a \right )^{4} b^{4}}+\frac {21 a \,c^{5} d^{2}}{2 \left (b x +a \right )^{4} b^{3}}-\frac {7 c^{6} d}{4 \left (b x +a \right )^{4} b^{2}}+\frac {7 a^{5} d^{7}}{\left (b x +a \right )^{3} b^{8}}-\frac {35 a^{4} c \,d^{6}}{\left (b x +a \right )^{3} b^{7}}+\frac {70 a^{3} c^{2} d^{5}}{\left (b x +a \right )^{3} b^{6}}-\frac {70 a^{2} c^{3} d^{4}}{\left (b x +a \right )^{3} b^{5}}+\frac {35 a \,c^{4} d^{3}}{\left (b x +a \right )^{3} b^{4}}-\frac {7 c^{5} d^{2}}{\left (b x +a \right )^{3} b^{3}}-\frac {35 a^{4} d^{7}}{2 \left (b x +a \right )^{2} b^{8}}+\frac {70 a^{3} c \,d^{6}}{\left (b x +a \right )^{2} b^{7}}-\frac {105 a^{2} c^{2} d^{5}}{\left (b x +a \right )^{2} b^{6}}+\frac {70 a \,c^{3} d^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {35 c^{4} d^{3}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {d^{7} x^{2}}{2 b^{6}}+\frac {35 a^{3} d^{7}}{\left (b x +a \right ) b^{8}}-\frac {105 a^{2} c \,d^{6}}{\left (b x +a \right ) b^{7}}+\frac {21 a^{2} d^{7} \ln \left (b x +a \right )}{b^{8}}+\frac {105 a \,c^{2} d^{5}}{\left (b x +a \right ) b^{6}}-\frac {42 a c \,d^{6} \ln \left (b x +a \right )}{b^{7}}-\frac {6 a \,d^{7} x}{b^{7}}-\frac {35 c^{3} d^{4}}{\left (b x +a \right ) b^{5}}+\frac {21 c^{2} d^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {7 c \,d^{6} x}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^6,x)

[Out]

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

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maxima [B]  time = 1.82, size = 504, normalized size = 2.78 \begin {gather*} -\frac {4 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 14 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} - 959 \, a^{5} b^{2} c^{2} d^{5} + 1218 \, a^{6} b c d^{6} - 459 \, a^{7} d^{7} + 700 \, {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 350 \, {\left (b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} - 18 \, a^{2} b^{5} c^{2} d^{5} + 20 \, a^{3} b^{4} c d^{6} - 7 \, a^{4} b^{3} d^{7}\right )} x^{3} + 70 \, {\left (2 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 110 \, a^{3} b^{4} c^{2} d^{5} + 130 \, a^{4} b^{3} c d^{6} - 47 \, a^{5} b^{2} d^{7}\right )} x^{2} + 35 \, {\left (b^{7} c^{6} d + 2 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} - 125 \, a^{4} b^{3} c^{2} d^{5} + 154 \, a^{5} b^{2} c d^{6} - 57 \, a^{6} b d^{7}\right )} x}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac {b d^{7} x^{2} + 2 \, {\left (7 \, b c d^{6} - 6 \, a d^{7}\right )} x}{2 \, b^{7}} + \frac {21 \, {\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} 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)^6,x, algorithm="maxima")

[Out]

-1/20*(4*b^7*c^7 + 7*a*b^6*c^6*d + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 - 959*a^5*b^2
*c^2*d^5 + 1218*a^6*b*c*d^6 - 459*a^7*d^7 + 700*(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 - a^3*b^4*d^7
)*x^4 + 350*(b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 - 18*a^2*b^5*c^2*d^5 + 20*a^3*b^4*c*d^6 - 7*a^4*b^3*d^7)*x^3 + 70*(
2*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 - 110*a^3*b^4*c^2*d^5 + 130*a^4*b^3*c*d^6 - 47*a^5*b^2*d^
7)*x^2 + 35*(b^7*c^6*d + 2*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 - 125*a^4*b^3*c^2*d^5 + 154*
a^5*b^2*c*d^6 - 57*a^6*b*d^7)*x)/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x +
a^5*b^8) + 1/2*(b*d^7*x^2 + 2*(7*b*c*d^6 - 6*a*d^7)*x)/b^7 + 21*(b^2*c^2*d^5 - 2*a*b*c*d^6 + a^2*d^7)*log(b*x
+ a)/b^8

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mupad [B]  time = 0.34, size = 508, normalized size = 2.81 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (21\,a^2\,d^7-42\,a\,b\,c\,d^6+21\,b^2\,c^2\,d^5\right )}{b^8}-x\,\left (\frac {6\,a\,d^7}{b^7}-\frac {7\,c\,d^6}{b^6}\right )-\frac {\frac {-459\,a^7\,d^7+1218\,a^6\,b\,c\,d^6-959\,a^5\,b^2\,c^2\,d^5+140\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+14\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+4\,b^7\,c^7}{20\,b}+x\,\left (-\frac {399\,a^6\,d^7}{4}+\frac {539\,a^5\,b\,c\,d^6}{2}-\frac {875\,a^4\,b^2\,c^2\,d^5}{4}+35\,a^3\,b^3\,c^3\,d^4+\frac {35\,a^2\,b^4\,c^4\,d^3}{4}+\frac {7\,a\,b^5\,c^5\,d^2}{2}+\frac {7\,b^6\,c^6\,d}{4}\right )+x^3\,\left (-\frac {245\,a^4\,b^2\,d^7}{2}+350\,a^3\,b^3\,c\,d^6-315\,a^2\,b^4\,c^2\,d^5+70\,a\,b^5\,c^3\,d^4+\frac {35\,b^6\,c^4\,d^3}{2}\right )+x^2\,\left (-\frac {329\,a^5\,b\,d^7}{2}+455\,a^4\,b^2\,c\,d^6-385\,a^3\,b^3\,c^2\,d^5+70\,a^2\,b^4\,c^3\,d^4+\frac {35\,a\,b^5\,c^4\,d^3}{2}+7\,b^6\,c^5\,d^2\right )-x^4\,\left (35\,a^3\,b^3\,d^7-105\,a^2\,b^4\,c\,d^6+105\,a\,b^5\,c^2\,d^5-35\,b^6\,c^3\,d^4\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}+\frac {d^7\,x^2}{2\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^6,x)

[Out]

(log(a + b*x)*(21*a^2*d^7 + 21*b^2*c^2*d^5 - 42*a*b*c*d^6))/b^8 - x*((6*a*d^7)/b^7 - (7*c*d^6)/b^6) - ((4*b^7*
c^7 - 459*a^7*d^7 + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 - 959*a^5*b^2*c^2*d^5 + 7*a*
b^6*c^6*d + 1218*a^6*b*c*d^6)/(20*b) + x*((7*b^6*c^6*d)/4 - (399*a^6*d^7)/4 + (7*a*b^5*c^5*d^2)/2 + (35*a^2*b^
4*c^4*d^3)/4 + 35*a^3*b^3*c^3*d^4 - (875*a^4*b^2*c^2*d^5)/4 + (539*a^5*b*c*d^6)/2) + x^3*((35*b^6*c^4*d^3)/2 -
(245*a^4*b^2*d^7)/2 + 70*a*b^5*c^3*d^4 + 350*a^3*b^3*c*d^6 - 315*a^2*b^4*c^2*d^5) + x^2*(7*b^6*c^5*d^2 - (329
*a^5*b*d^7)/2 + (35*a*b^5*c^4*d^3)/2 + 455*a^4*b^2*c*d^6 + 70*a^2*b^4*c^3*d^4 - 385*a^3*b^3*c^2*d^5) - x^4*(35
*a^3*b^3*d^7 - 35*b^6*c^3*d^4 + 105*a*b^5*c^2*d^5 - 105*a^2*b^4*c*d^6))/(a^5*b^7 + b^12*x^5 + 5*a^4*b^8*x + 5*
a*b^11*x^4 + 10*a^3*b^9*x^2 + 10*a^2*b^10*x^3) + (d^7*x^2)/(2*b^6)

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sympy [B]  time = 97.19, size = 524, normalized size = 2.90 \begin {gather*} x \left (- \frac {6 a d^{7}}{b^{7}} + \frac {7 c d^{6}}{b^{6}}\right ) + \frac {459 a^{7} d^{7} - 1218 a^{6} b c d^{6} + 959 a^{5} b^{2} c^{2} d^{5} - 140 a^{4} b^{3} c^{3} d^{4} - 35 a^{3} b^{4} c^{4} d^{3} - 14 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 4 b^{7} c^{7} + x^{4} \left (700 a^{3} b^{4} d^{7} - 2100 a^{2} b^{5} c d^{6} + 2100 a b^{6} c^{2} d^{5} - 700 b^{7} c^{3} d^{4}\right ) + x^{3} \left (2450 a^{4} b^{3} d^{7} - 7000 a^{3} b^{4} c d^{6} + 6300 a^{2} b^{5} c^{2} d^{5} - 1400 a b^{6} c^{3} d^{4} - 350 b^{7} c^{4} d^{3}\right ) + x^{2} \left (3290 a^{5} b^{2} d^{7} - 9100 a^{4} b^{3} c d^{6} + 7700 a^{3} b^{4} c^{2} d^{5} - 1400 a^{2} b^{5} c^{3} d^{4} - 350 a b^{6} c^{4} d^{3} - 140 b^{7} c^{5} d^{2}\right ) + x \left (1995 a^{6} b d^{7} - 5390 a^{5} b^{2} c d^{6} + 4375 a^{4} b^{3} c^{2} d^{5} - 700 a^{3} b^{4} c^{3} d^{4} - 175 a^{2} b^{5} c^{4} d^{3} - 70 a b^{6} c^{5} d^{2} - 35 b^{7} c^{6} d\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} + \frac {d^{7} x^{2}}{2 b^{6}} + \frac {21 d^{5} \left (a d - b c\right )^{2} \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)**6,x)

[Out]

x*(-6*a*d**7/b**7 + 7*c*d**6/b**6) + (459*a**7*d**7 - 1218*a**6*b*c*d**6 + 959*a**5*b**2*c**2*d**5 - 140*a**4*
b**3*c**3*d**4 - 35*a**3*b**4*c**4*d**3 - 14*a**2*b**5*c**5*d**2 - 7*a*b**6*c**6*d - 4*b**7*c**7 + x**4*(700*a
**3*b**4*d**7 - 2100*a**2*b**5*c*d**6 + 2100*a*b**6*c**2*d**5 - 700*b**7*c**3*d**4) + x**3*(2450*a**4*b**3*d**
7 - 7000*a**3*b**4*c*d**6 + 6300*a**2*b**5*c**2*d**5 - 1400*a*b**6*c**3*d**4 - 350*b**7*c**4*d**3) + x**2*(329
0*a**5*b**2*d**7 - 9100*a**4*b**3*c*d**6 + 7700*a**3*b**4*c**2*d**5 - 1400*a**2*b**5*c**3*d**4 - 350*a*b**6*c*
*4*d**3 - 140*b**7*c**5*d**2) + x*(1995*a**6*b*d**7 - 5390*a**5*b**2*c*d**6 + 4375*a**4*b**3*c**2*d**5 - 700*a
**3*b**4*c**3*d**4 - 175*a**2*b**5*c**4*d**3 - 70*a*b**6*c**5*d**2 - 35*b**7*c**6*d))/(20*a**5*b**8 + 100*a**4
*b**9*x + 200*a**3*b**10*x**2 + 200*a**2*b**11*x**3 + 100*a*b**12*x**4 + 20*b**13*x**5) + d**7*x**2/(2*b**6) +
21*d**5*(a*d - b*c)**2*log(a + b*x)/b**8

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