3.12.79 \(\int \frac {(c+d x)^7}{(a+b x)^3} \, dx\)

Optimal. Leaf size=185 \[ \frac {7 d^6 (a+b x)^4 (b c-a d)}{4 b^8}+\frac {7 d^5 (a+b x)^3 (b c-a d)^2}{b^8}+\frac {35 d^4 (a+b x)^2 (b c-a d)^3}{2 b^8}+\frac {21 d^2 (b c-a d)^5 \log (a+b x)}{b^8}-\frac {7 d (b c-a d)^6}{b^8 (a+b x)}-\frac {(b c-a d)^7}{2 b^8 (a+b x)^2}+\frac {d^7 (a+b x)^5}{5 b^8}+\frac {35 d^3 x (b c-a d)^4}{b^7} \]

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Rubi [A]  time = 0.22, antiderivative size = 185, 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)^4 (b c-a d)}{4 b^8}+\frac {7 d^5 (a+b x)^3 (b c-a d)^2}{b^8}+\frac {35 d^4 (a+b x)^2 (b c-a d)^3}{2 b^8}+\frac {35 d^3 x (b c-a d)^4}{b^7}+\frac {21 d^2 (b c-a d)^5 \log (a+b x)}{b^8}-\frac {7 d (b c-a d)^6}{b^8 (a+b x)}-\frac {(b c-a d)^7}{2 b^8 (a+b x)^2}+\frac {d^7 (a+b x)^5}{5 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^3,x]

[Out]

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

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Mathematica [B]  time = 0.13, size = 389, normalized size = 2.10 \begin {gather*} \frac {-130 a^7 d^7+10 a^6 b d^6 (77 c+16 d x)+10 a^5 b^2 d^5 \left (-189 c^2-56 c d x+50 d^2 x^2\right )+70 a^4 b^3 d^4 \left (35 c^3+6 c^2 d x-34 c d^2 x^2+2 d^3 x^3\right )-35 a^3 b^4 d^3 \left (50 c^4-20 c^3 d x-126 c^2 d^2 x^2+20 c d^3 x^3+d^4 x^4\right )+7 a^2 b^5 d^2 \left (90 c^5-200 c^4 d x-550 c^3 d^2 x^2+200 c^2 d^3 x^3+25 c d^4 x^4+2 d^5 x^5\right )-7 a b^6 d \left (10 c^6-120 c^5 d x-200 c^4 d^2 x^2+200 c^3 d^3 x^3+50 c^2 d^4 x^4+10 c d^5 x^5+d^6 x^6\right )-420 d^2 (a+b x)^2 (a d-b c)^5 \log (a+b x)+b^7 \left (-10 c^7-140 c^6 d x+700 c^4 d^3 x^3+350 c^3 d^4 x^4+140 c^2 d^5 x^5+35 c d^6 x^6+4 d^7 x^7\right )}{20 b^8 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^3,x]

[Out]

(-130*a^7*d^7 + 10*a^6*b*d^6*(77*c + 16*d*x) + 10*a^5*b^2*d^5*(-189*c^2 - 56*c*d*x + 50*d^2*x^2) + 70*a^4*b^3*
d^4*(35*c^3 + 6*c^2*d*x - 34*c*d^2*x^2 + 2*d^3*x^3) - 35*a^3*b^4*d^3*(50*c^4 - 20*c^3*d*x - 126*c^2*d^2*x^2 +
20*c*d^3*x^3 + d^4*x^4) + 7*a^2*b^5*d^2*(90*c^5 - 200*c^4*d*x - 550*c^3*d^2*x^2 + 200*c^2*d^3*x^3 + 25*c*d^4*x
^4 + 2*d^5*x^5) - 7*a*b^6*d*(10*c^6 - 120*c^5*d*x - 200*c^4*d^2*x^2 + 200*c^3*d^3*x^3 + 50*c^2*d^4*x^4 + 10*c*
d^5*x^5 + d^6*x^6) + b^7*(-10*c^7 - 140*c^6*d*x + 700*c^4*d^3*x^3 + 350*c^3*d^4*x^4 + 140*c^2*d^5*x^5 + 35*c*d
^6*x^6 + 4*d^7*x^7) - 420*d^2*(-(b*c) + a*d)^5*(a + b*x)^2*Log[a + b*x])/(20*b^8*(a + b*x)^2)

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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)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^3,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^3, x]

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fricas [B]  time = 1.20, size = 703, normalized size = 3.80 \begin {gather*} \frac {4 \, b^{7} d^{7} x^{7} - 10 \, b^{7} c^{7} - 70 \, a b^{6} c^{6} d + 630 \, a^{2} b^{5} c^{5} d^{2} - 1750 \, a^{3} b^{4} c^{4} d^{3} + 2450 \, a^{4} b^{3} c^{3} d^{4} - 1890 \, a^{5} b^{2} c^{2} d^{5} + 770 \, a^{6} b c d^{6} - 130 \, a^{7} d^{7} + 7 \, {\left (5 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 14 \, {\left (10 \, b^{7} c^{2} d^{5} - 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 35 \, {\left (10 \, b^{7} c^{3} d^{4} - 10 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 140 \, {\left (5 \, b^{7} c^{4} d^{3} - 10 \, a b^{6} c^{3} d^{4} + 10 \, a^{2} b^{5} c^{2} d^{5} - 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 10 \, {\left (140 \, a b^{6} c^{4} d^{3} - 385 \, a^{2} b^{5} c^{3} d^{4} + 441 \, a^{3} b^{4} c^{2} d^{5} - 238 \, a^{4} b^{3} c d^{6} + 50 \, a^{5} b^{2} d^{7}\right )} x^{2} - 20 \, {\left (7 \, b^{7} c^{6} d - 42 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} - 35 \, a^{3} b^{4} c^{3} d^{4} - 21 \, a^{4} b^{3} c^{2} d^{5} + 28 \, a^{5} b^{2} c d^{6} - 8 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{2} b^{5} c^{5} d^{2} - 5 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} - 10 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{5} d^{2} - 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} - 10 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{20 \, {\left (b^{10} x^{2} + 2 \, a b^{9} x + a^{2} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*b^7*d^7*x^7 - 10*b^7*c^7 - 70*a*b^6*c^6*d + 630*a^2*b^5*c^5*d^2 - 1750*a^3*b^4*c^4*d^3 + 2450*a^4*b^3*
c^3*d^4 - 1890*a^5*b^2*c^2*d^5 + 770*a^6*b*c*d^6 - 130*a^7*d^7 + 7*(5*b^7*c*d^6 - a*b^6*d^7)*x^6 + 14*(10*b^7*
c^2*d^5 - 5*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 35*(10*b^7*c^3*d^4 - 10*a*b^6*c^2*d^5 + 5*a^2*b^5*c*d^6 - a^3*b^4
*d^7)*x^4 + 140*(5*b^7*c^4*d^3 - 10*a*b^6*c^3*d^4 + 10*a^2*b^5*c^2*d^5 - 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 +
10*(140*a*b^6*c^4*d^3 - 385*a^2*b^5*c^3*d^4 + 441*a^3*b^4*c^2*d^5 - 238*a^4*b^3*c*d^6 + 50*a^5*b^2*d^7)*x^2 -
20*(7*b^7*c^6*d - 42*a*b^6*c^5*d^2 + 70*a^2*b^5*c^4*d^3 - 35*a^3*b^4*c^3*d^4 - 21*a^4*b^3*c^2*d^5 + 28*a^5*b^2
*c*d^6 - 8*a^6*b*d^7)*x + 420*(a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4 - 10*a^5*b^2*c^2*d^5 +
5*a^6*b*c*d^6 - a^7*d^7 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^
3*c*d^6 - a^5*b^2*d^7)*x^2 + 2*(a*b^6*c^5*d^2 - 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 - 10*a^4*b^3*c^2*d^5 +
5*a^5*b^2*c*d^6 - a^6*b*d^7)*x)*log(b*x + a))/(b^10*x^2 + 2*a*b^9*x + a^2*b^8)

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giac [B]  time = 1.26, size = 477, normalized size = 2.58 \begin {gather*} \frac {21 \, {\left (b^{5} c^{5} d^{2} - 5 \, a b^{4} c^{4} d^{3} + 10 \, a^{2} b^{3} c^{3} d^{4} - 10 \, a^{3} b^{2} c^{2} d^{5} + 5 \, a^{4} b c d^{6} - a^{5} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {b^{7} c^{7} + 7 \, a b^{6} c^{6} d - 63 \, a^{2} b^{5} c^{5} d^{2} + 175 \, a^{3} b^{4} c^{4} d^{3} - 245 \, a^{4} b^{3} c^{3} d^{4} + 189 \, a^{5} b^{2} c^{2} d^{5} - 77 \, a^{6} b c d^{6} + 13 \, a^{7} d^{7} + 14 \, {\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}{2 \, {\left (b x + a\right )}^{2} b^{8}} + \frac {4 \, b^{12} d^{7} x^{5} + 35 \, b^{12} c d^{6} x^{4} - 15 \, a b^{11} d^{7} x^{4} + 140 \, b^{12} c^{2} d^{5} x^{3} - 140 \, a b^{11} c d^{6} x^{3} + 40 \, a^{2} b^{10} d^{7} x^{3} + 350 \, b^{12} c^{3} d^{4} x^{2} - 630 \, a b^{11} c^{2} d^{5} x^{2} + 420 \, a^{2} b^{10} c d^{6} x^{2} - 100 \, a^{3} b^{9} d^{7} x^{2} + 700 \, b^{12} c^{4} d^{3} x - 2100 \, a b^{11} c^{3} d^{4} x + 2520 \, a^{2} b^{10} c^{2} d^{5} x - 1400 \, a^{3} b^{9} c d^{6} x + 300 \, a^{4} b^{8} d^{7} x}{20 \, b^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21*(b^5*c^5*d^2 - 5*a*b^4*c^4*d^3 + 10*a^2*b^3*c^3*d^4 - 10*a^3*b^2*c^2*d^5 + 5*a^4*b*c*d^6 - a^5*d^7)*log(abs
(b*x + a))/b^8 - 1/2*(b^7*c^7 + 7*a*b^6*c^6*d - 63*a^2*b^5*c^5*d^2 + 175*a^3*b^4*c^4*d^3 - 245*a^4*b^3*c^3*d^4
+ 189*a^5*b^2*c^2*d^5 - 77*a^6*b*c*d^6 + 13*a^7*d^7 + 14*(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)/((b*x + a)^2*b^8) + 1/20*(4*b^12*d^7
*x^5 + 35*b^12*c*d^6*x^4 - 15*a*b^11*d^7*x^4 + 140*b^12*c^2*d^5*x^3 - 140*a*b^11*c*d^6*x^3 + 40*a^2*b^10*d^7*x
^3 + 350*b^12*c^3*d^4*x^2 - 630*a*b^11*c^2*d^5*x^2 + 420*a^2*b^10*c*d^6*x^2 - 100*a^3*b^9*d^7*x^2 + 700*b^12*c
^4*d^3*x - 2100*a*b^11*c^3*d^4*x + 2520*a^2*b^10*c^2*d^5*x - 1400*a^3*b^9*c*d^6*x + 300*a^4*b^8*d^7*x)/b^15

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maple [B]  time = 0.01, size = 599, normalized size = 3.24 \begin {gather*} \frac {d^{7} x^{5}}{5 b^{3}}-\frac {3 a \,d^{7} x^{4}}{4 b^{4}}+\frac {7 c \,d^{6} x^{4}}{4 b^{3}}+\frac {2 a^{2} d^{7} x^{3}}{b^{5}}-\frac {7 a c \,d^{6} x^{3}}{b^{4}}+\frac {7 c^{2} d^{5} x^{3}}{b^{3}}+\frac {a^{7} d^{7}}{2 \left (b x +a \right )^{2} b^{8}}-\frac {7 a^{6} c \,d^{6}}{2 \left (b x +a \right )^{2} b^{7}}+\frac {21 a^{5} c^{2} d^{5}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {35 a^{4} c^{3} d^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {35 a^{3} c^{4} d^{3}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {5 a^{3} d^{7} x^{2}}{b^{6}}-\frac {21 a^{2} c^{5} d^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {21 a^{2} c \,d^{6} x^{2}}{b^{5}}+\frac {7 a \,c^{6} d}{2 \left (b x +a \right )^{2} b^{2}}-\frac {63 a \,c^{2} d^{5} x^{2}}{2 b^{4}}-\frac {c^{7}}{2 \left (b x +a \right )^{2} b}+\frac {35 c^{3} d^{4} x^{2}}{2 b^{3}}-\frac {7 a^{6} d^{7}}{\left (b x +a \right ) b^{8}}+\frac {42 a^{5} c \,d^{6}}{\left (b x +a \right ) b^{7}}-\frac {21 a^{5} d^{7} \ln \left (b x +a \right )}{b^{8}}-\frac {105 a^{4} c^{2} d^{5}}{\left (b x +a \right ) b^{6}}+\frac {105 a^{4} c \,d^{6} \ln \left (b x +a \right )}{b^{7}}+\frac {15 a^{4} d^{7} x}{b^{7}}+\frac {140 a^{3} c^{3} d^{4}}{\left (b x +a \right ) b^{5}}-\frac {210 a^{3} c^{2} d^{5} \ln \left (b x +a \right )}{b^{6}}-\frac {70 a^{3} c \,d^{6} x}{b^{6}}-\frac {105 a^{2} c^{4} d^{3}}{\left (b x +a \right ) b^{4}}+\frac {210 a^{2} c^{3} d^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {126 a^{2} c^{2} d^{5} x}{b^{5}}+\frac {42 a \,c^{5} d^{2}}{\left (b x +a \right ) b^{3}}-\frac {105 a \,c^{4} d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {105 a \,c^{3} d^{4} x}{b^{4}}-\frac {7 c^{6} d}{\left (b x +a \right ) b^{2}}+\frac {21 c^{5} d^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {35 c^{4} d^{3} x}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^7/b^5*x^3*a^2+7*d^5/b^3*x^3*c^2-5*d^7/b^6*x^2*a^3+35/2*d^4/b^3*x^2*c^3+15*d^7/b^7*a^4*x+35*d^3/b^3*c^4*x+1
/2/b^8/(b*x+a)^2*a^7*d^7-21/b^8*d^7*ln(b*x+a)*a^5+21/b^3*d^2*ln(b*x+a)*c^5-7/b^8*d^7/(b*x+a)*a^6-7/b^2*d/(b*x+
a)*c^6-3/4*d^7/b^4*x^4*a+7/4*d^6/b^3*x^4*c+210/b^5*d^4*ln(b*x+a)*a^2*c^3-105/b^4*d^3*ln(b*x+a)*a*c^4+42/b^7*d^
6/(b*x+a)*a^5*c-105/b^6*d^5/(b*x+a)*a^4*c^2+140/b^5*d^4/(b*x+a)*a^3*c^3-105/b^4*d^3/(b*x+a)*a^2*c^4+42/b^3*d^2
/(b*x+a)*a*c^5-63/2*d^5/b^4*x^2*a*c^2-70*d^6/b^6*a^3*c*x+126*d^5/b^5*a^2*c^2*x-105*d^4/b^4*a*c^3*x-7/2/b^7/(b*
x+a)^2*a^6*c*d^6+21/2/b^6/(b*x+a)^2*a^5*c^2*d^5-35/2/b^5/(b*x+a)^2*a^4*c^3*d^4+35/2/b^4/(b*x+a)^2*a^3*c^4*d^3-
21/2/b^3/(b*x+a)^2*a^2*c^5*d^2+7/2/b^2/(b*x+a)^2*a*c^6*d+105/b^7*d^6*ln(b*x+a)*a^4*c-210/b^6*d^5*ln(b*x+a)*a^3
*c^2-7*d^6/b^4*x^3*a*c+21*d^6/b^5*x^2*a^2*c-1/2/b/(b*x+a)^2*c^7+1/5*d^7/b^3*x^5

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maxima [B]  time = 1.54, size = 473, normalized size = 2.56 \begin {gather*} -\frac {b^{7} c^{7} + 7 \, a b^{6} c^{6} d - 63 \, a^{2} b^{5} c^{5} d^{2} + 175 \, a^{3} b^{4} c^{4} d^{3} - 245 \, a^{4} b^{3} c^{3} d^{4} + 189 \, a^{5} b^{2} c^{2} d^{5} - 77 \, a^{6} b c d^{6} + 13 \, a^{7} d^{7} + 14 \, {\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}{2 \, {\left (b^{10} x^{2} + 2 \, a b^{9} x + a^{2} b^{8}\right )}} + \frac {4 \, b^{4} d^{7} x^{5} + 5 \, {\left (7 \, b^{4} c d^{6} - 3 \, a b^{3} d^{7}\right )} x^{4} + 20 \, {\left (7 \, b^{4} c^{2} d^{5} - 7 \, a b^{3} c d^{6} + 2 \, a^{2} b^{2} d^{7}\right )} x^{3} + 10 \, {\left (35 \, b^{4} c^{3} d^{4} - 63 \, a b^{3} c^{2} d^{5} + 42 \, a^{2} b^{2} c d^{6} - 10 \, a^{3} b d^{7}\right )} x^{2} + 20 \, {\left (35 \, b^{4} c^{4} d^{3} - 105 \, a b^{3} c^{3} d^{4} + 126 \, a^{2} b^{2} c^{2} d^{5} - 70 \, a^{3} b c d^{6} + 15 \, a^{4} d^{7}\right )} x}{20 \, b^{7}} + \frac {21 \, {\left (b^{5} c^{5} d^{2} - 5 \, a b^{4} c^{4} d^{3} + 10 \, a^{2} b^{3} c^{3} d^{4} - 10 \, a^{3} b^{2} c^{2} d^{5} + 5 \, a^{4} b c d^{6} - a^{5} 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)^3,x, algorithm="maxima")

[Out]

-1/2*(b^7*c^7 + 7*a*b^6*c^6*d - 63*a^2*b^5*c^5*d^2 + 175*a^3*b^4*c^4*d^3 - 245*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c
^2*d^5 - 77*a^6*b*c*d^6 + 13*a^7*d^7 + 14*(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)/(b^10*x^2 + 2*a*b^9*x + a^2*b^8) + 1/20*(4*b^4*d^7*x
^5 + 5*(7*b^4*c*d^6 - 3*a*b^3*d^7)*x^4 + 20*(7*b^4*c^2*d^5 - 7*a*b^3*c*d^6 + 2*a^2*b^2*d^7)*x^3 + 10*(35*b^4*c
^3*d^4 - 63*a*b^3*c^2*d^5 + 42*a^2*b^2*c*d^6 - 10*a^3*b*d^7)*x^2 + 20*(35*b^4*c^4*d^3 - 105*a*b^3*c^3*d^4 + 12
6*a^2*b^2*c^2*d^5 - 70*a^3*b*c*d^6 + 15*a^4*d^7)*x)/b^7 + 21*(b^5*c^5*d^2 - 5*a*b^4*c^4*d^3 + 10*a^2*b^3*c^3*d
^4 - 10*a^3*b^2*c^2*d^5 + 5*a^4*b*c*d^6 - a^5*d^7)*log(b*x + a)/b^8

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mupad [B]  time = 0.27, size = 690, normalized size = 3.73 \begin {gather*} x\,\left (\frac {3\,a\,\left (\frac {3\,a\,\left (\frac {3\,a\,\left (\frac {3\,a\,d^7}{b^4}-\frac {7\,c\,d^6}{b^3}\right )}{b}-\frac {3\,a^2\,d^7}{b^5}+\frac {21\,c^2\,d^5}{b^3}\right )}{b}+\frac {a^3\,d^7}{b^6}-\frac {35\,c^3\,d^4}{b^3}-\frac {3\,a^2\,\left (\frac {3\,a\,d^7}{b^4}-\frac {7\,c\,d^6}{b^3}\right )}{b^2}\right )}{b}+\frac {35\,c^4\,d^3}{b^3}+\frac {a^3\,\left (\frac {3\,a\,d^7}{b^4}-\frac {7\,c\,d^6}{b^3}\right )}{b^3}-\frac {3\,a^2\,\left (\frac {3\,a\,\left (\frac {3\,a\,d^7}{b^4}-\frac {7\,c\,d^6}{b^3}\right )}{b}-\frac {3\,a^2\,d^7}{b^5}+\frac {21\,c^2\,d^5}{b^3}\right )}{b^2}\right )-x^4\,\left (\frac {3\,a\,d^7}{4\,b^4}-\frac {7\,c\,d^6}{4\,b^3}\right )-x^2\,\left (\frac {3\,a\,\left (\frac {3\,a\,\left (\frac {3\,a\,d^7}{b^4}-\frac {7\,c\,d^6}{b^3}\right )}{b}-\frac {3\,a^2\,d^7}{b^5}+\frac {21\,c^2\,d^5}{b^3}\right )}{2\,b}+\frac {a^3\,d^7}{2\,b^6}-\frac {35\,c^3\,d^4}{2\,b^3}-\frac {3\,a^2\,\left (\frac {3\,a\,d^7}{b^4}-\frac {7\,c\,d^6}{b^3}\right )}{2\,b^2}\right )+x^3\,\left (\frac {a\,\left (\frac {3\,a\,d^7}{b^4}-\frac {7\,c\,d^6}{b^3}\right )}{b}-\frac {a^2\,d^7}{b^5}+\frac {7\,c^2\,d^5}{b^3}\right )-\frac {\frac {13\,a^7\,d^7-77\,a^6\,b\,c\,d^6+189\,a^5\,b^2\,c^2\,d^5-245\,a^4\,b^3\,c^3\,d^4+175\,a^3\,b^4\,c^4\,d^3-63\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+b^7\,c^7}{2\,b}+x\,\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 )}{a^2\,b^7+2\,a\,b^8\,x+b^9\,x^2}+\frac {d^7\,x^5}{5\,b^3}-\frac {\ln \left (a+b\,x\right )\,\left (21\,a^5\,d^7-105\,a^4\,b\,c\,d^6+210\,a^3\,b^2\,c^2\,d^5-210\,a^2\,b^3\,c^3\,d^4+105\,a\,b^4\,c^4\,d^3-21\,b^5\,c^5\,d^2\right )}{b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^3,x)

[Out]

x*((3*a*((3*a*((3*a*((3*a*d^7)/b^4 - (7*c*d^6)/b^3))/b - (3*a^2*d^7)/b^5 + (21*c^2*d^5)/b^3))/b + (a^3*d^7)/b^
6 - (35*c^3*d^4)/b^3 - (3*a^2*((3*a*d^7)/b^4 - (7*c*d^6)/b^3))/b^2))/b + (35*c^4*d^3)/b^3 + (a^3*((3*a*d^7)/b^
4 - (7*c*d^6)/b^3))/b^3 - (3*a^2*((3*a*((3*a*d^7)/b^4 - (7*c*d^6)/b^3))/b - (3*a^2*d^7)/b^5 + (21*c^2*d^5)/b^3
))/b^2) - x^4*((3*a*d^7)/(4*b^4) - (7*c*d^6)/(4*b^3)) - x^2*((3*a*((3*a*((3*a*d^7)/b^4 - (7*c*d^6)/b^3))/b - (
3*a^2*d^7)/b^5 + (21*c^2*d^5)/b^3))/(2*b) + (a^3*d^7)/(2*b^6) - (35*c^3*d^4)/(2*b^3) - (3*a^2*((3*a*d^7)/b^4 -
(7*c*d^6)/b^3))/(2*b^2)) + x^3*((a*((3*a*d^7)/b^4 - (7*c*d^6)/b^3))/b - (a^2*d^7)/b^5 + (7*c^2*d^5)/b^3) - ((
13*a^7*d^7 + b^7*c^7 - 63*a^2*b^5*c^5*d^2 + 175*a^3*b^4*c^4*d^3 - 245*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2*d^5 +
7*a*b^6*c^6*d - 77*a^6*b*c*d^6)/(2*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))/(a^2*b^7 + b^9*x^2 + 2*a*b^8*x) + (d^7*x^5)/(5*b^
3) - (log(a + b*x)*(21*a^5*d^7 - 21*b^5*c^5*d^2 + 105*a*b^4*c^4*d^3 - 210*a^2*b^3*c^3*d^4 + 210*a^3*b^2*c^2*d^
5 - 105*a^4*b*c*d^6))/b^8

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sympy [B]  time = 2.95, size = 447, normalized size = 2.42 \begin {gather*} x^{4} \left (- \frac {3 a d^{7}}{4 b^{4}} + \frac {7 c d^{6}}{4 b^{3}}\right ) + x^{3} \left (\frac {2 a^{2} d^{7}}{b^{5}} - \frac {7 a c d^{6}}{b^{4}} + \frac {7 c^{2} d^{5}}{b^{3}}\right ) + x^{2} \left (- \frac {5 a^{3} d^{7}}{b^{6}} + \frac {21 a^{2} c d^{6}}{b^{5}} - \frac {63 a c^{2} d^{5}}{2 b^{4}} + \frac {35 c^{3} d^{4}}{2 b^{3}}\right ) + x \left (\frac {15 a^{4} d^{7}}{b^{7}} - \frac {70 a^{3} c d^{6}}{b^{6}} + \frac {126 a^{2} c^{2} d^{5}}{b^{5}} - \frac {105 a c^{3} d^{4}}{b^{4}} + \frac {35 c^{4} d^{3}}{b^{3}}\right ) + \frac {- 13 a^{7} d^{7} + 77 a^{6} b c d^{6} - 189 a^{5} b^{2} c^{2} d^{5} + 245 a^{4} b^{3} c^{3} d^{4} - 175 a^{3} b^{4} c^{4} d^{3} + 63 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - b^{7} c^{7} + x \left (- 14 a^{6} b d^{7} + 84 a^{5} b^{2} c d^{6} - 210 a^{4} b^{3} c^{2} d^{5} + 280 a^{3} b^{4} c^{3} d^{4} - 210 a^{2} b^{5} c^{4} d^{3} + 84 a b^{6} c^{5} d^{2} - 14 b^{7} c^{6} d\right )}{2 a^{2} b^{8} + 4 a b^{9} x + 2 b^{10} x^{2}} + \frac {d^{7} x^{5}}{5 b^{3}} - \frac {21 d^{2} \left (a d - b c\right )^{5} \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)**3,x)

[Out]

x**4*(-3*a*d**7/(4*b**4) + 7*c*d**6/(4*b**3)) + x**3*(2*a**2*d**7/b**5 - 7*a*c*d**6/b**4 + 7*c**2*d**5/b**3) +
x**2*(-5*a**3*d**7/b**6 + 21*a**2*c*d**6/b**5 - 63*a*c**2*d**5/(2*b**4) + 35*c**3*d**4/(2*b**3)) + x*(15*a**4
*d**7/b**7 - 70*a**3*c*d**6/b**6 + 126*a**2*c**2*d**5/b**5 - 105*a*c**3*d**4/b**4 + 35*c**4*d**3/b**3) + (-13*
a**7*d**7 + 77*a**6*b*c*d**6 - 189*a**5*b**2*c**2*d**5 + 245*a**4*b**3*c**3*d**4 - 175*a**3*b**4*c**4*d**3 + 6
3*a**2*b**5*c**5*d**2 - 7*a*b**6*c**6*d - b**7*c**7 + x*(-14*a**6*b*d**7 + 84*a**5*b**2*c*d**6 - 210*a**4*b**3
*c**2*d**5 + 280*a**3*b**4*c**3*d**4 - 210*a**2*b**5*c**4*d**3 + 84*a*b**6*c**5*d**2 - 14*b**7*c**6*d))/(2*a**
2*b**8 + 4*a*b**9*x + 2*b**10*x**2) + d**7*x**5/(5*b**3) - 21*d**2*(a*d - b*c)**5*log(a + b*x)/b**8

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