3.12.80 \(\int \frac {(c+d x)^7}{(a+b x)^4} \, dx\)

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

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Rubi [A]  time = 0.21, 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)^3 (b c-a d)}{3 b^8}+\frac {21 d^5 (a+b x)^2 (b c-a d)^2}{2 b^8}+\frac {35 d^4 x (b c-a d)^3}{b^7}-\frac {21 d^2 (b c-a d)^5}{b^8 (a+b x)}+\frac {35 d^3 (b c-a d)^4 \log (a+b x)}{b^8}-\frac {7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac {(b c-a d)^7}{3 b^8 (a+b x)^3}+\frac {d^7 (a+b x)^4}{4 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x)^4,x]

[Out]

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

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Mathematica [A]  time = 0.11, size = 199, normalized size = 1.06 \begin {gather*} \frac {6 b^2 d^5 x^2 \left (10 a^2 d^2-28 a b c d+21 b^2 c^2\right )+12 b d^4 x \left (-20 a^3 d^3+70 a^2 b c d^2-84 a b^2 c^2 d+35 b^3 c^3\right )+4 b^3 d^6 x^3 (7 b c-4 a d)+420 d^3 (b c-a d)^4 \log (a+b x)+\frac {252 d^2 (a d-b c)^5}{a+b x}-\frac {42 d (b c-a d)^6}{(a+b x)^2}-\frac {4 (b c-a d)^7}{(a+b x)^3}+3 b^4 d^7 x^4}{12 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x)^4,x]

[Out]

(12*b*d^4*(35*b^3*c^3 - 84*a*b^2*c^2*d + 70*a^2*b*c*d^2 - 20*a^3*d^3)*x + 6*b^2*d^5*(21*b^2*c^2 - 28*a*b*c*d +
10*a^2*d^2)*x^2 + 4*b^3*d^6*(7*b*c - 4*a*d)*x^3 + 3*b^4*d^7*x^4 - (4*(b*c - a*d)^7)/(a + b*x)^3 - (42*d*(b*c
- a*d)^6)/(a + b*x)^2 + (252*d^2*(-(b*c) + a*d)^5)/(a + b*x) + 420*d^3*(b*c - a*d)^4*Log[a + b*x])/(12*b^8)

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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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^4,x]

[Out]

IntegrateAlgebraic[(c + d*x)^7/(a + b*x)^4, x]

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fricas [B]  time = 1.51, size = 739, normalized size = 3.95 \begin {gather*} \frac {3 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 84 \, a^{2} b^{5} c^{5} d^{2} + 770 \, a^{3} b^{4} c^{4} d^{3} - 1820 \, a^{4} b^{3} c^{3} d^{4} + 1974 \, a^{5} b^{2} c^{2} d^{5} - 1036 \, a^{6} b c d^{6} + 214 \, a^{7} d^{7} + 7 \, {\left (4 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 21 \, {\left (6 \, b^{7} c^{2} d^{5} - 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \, {\left (4 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 2 \, {\left (630 \, a b^{6} c^{3} d^{4} - 1323 \, a^{2} b^{5} c^{2} d^{5} + 1022 \, a^{3} b^{4} c d^{6} - 278 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \, {\left (42 \, b^{7} c^{5} d^{2} - 210 \, a b^{6} c^{4} d^{3} + 210 \, a^{2} b^{5} c^{3} d^{4} + 63 \, a^{3} b^{4} c^{2} d^{5} - 182 \, a^{4} b^{3} c d^{6} + 68 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \, {\left (7 \, b^{7} c^{6} d + 42 \, a b^{6} c^{5} d^{2} - 315 \, a^{2} b^{5} c^{4} d^{3} + 630 \, a^{3} b^{4} c^{3} d^{4} - 567 \, a^{4} b^{3} c^{2} d^{5} + 238 \, a^{5} b^{2} c d^{6} - 37 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{3} b^{4} c^{4} d^{3} - 4 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} - 4 \, a^{6} b c d^{6} + a^{7} d^{7} + {\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 3 \, {\left (a b^{6} c^{4} d^{3} - 4 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} - 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} d^{3} - 4 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} - 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/12*(3*b^7*d^7*x^7 - 4*b^7*c^7 - 14*a*b^6*c^6*d - 84*a^2*b^5*c^5*d^2 + 770*a^3*b^4*c^4*d^3 - 1820*a^4*b^3*c^3
*d^4 + 1974*a^5*b^2*c^2*d^5 - 1036*a^6*b*c*d^6 + 214*a^7*d^7 + 7*(4*b^7*c*d^6 - a*b^6*d^7)*x^6 + 21*(6*b^7*c^2
*d^5 - 4*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 105*(4*b^7*c^3*d^4 - 6*a*b^6*c^2*d^5 + 4*a^2*b^5*c*d^6 - a^3*b^4*d^7
)*x^4 + 2*(630*a*b^6*c^3*d^4 - 1323*a^2*b^5*c^2*d^5 + 1022*a^3*b^4*c*d^6 - 278*a^4*b^3*d^7)*x^3 - 6*(42*b^7*c^
5*d^2 - 210*a*b^6*c^4*d^3 + 210*a^2*b^5*c^3*d^4 + 63*a^3*b^4*c^2*d^5 - 182*a^4*b^3*c*d^6 + 68*a^5*b^2*d^7)*x^2
- 6*(7*b^7*c^6*d + 42*a*b^6*c^5*d^2 - 315*a^2*b^5*c^4*d^3 + 630*a^3*b^4*c^3*d^4 - 567*a^4*b^3*c^2*d^5 + 238*a
^5*b^2*c*d^6 - 37*a^6*b*d^7)*x + 420*(a^3*b^4*c^4*d^3 - 4*a^4*b^3*c^3*d^4 + 6*a^5*b^2*c^2*d^5 - 4*a^6*b*c*d^6
+ a^7*d^7 + (b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 3*(a*b^6
*c^4*d^3 - 4*a^2*b^5*c^3*d^4 + 6*a^3*b^4*c^2*d^5 - 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 3*(a^2*b^5*c^4*d^3 - 4
*a^3*b^4*c^3*d^4 + 6*a^4*b^3*c^2*d^5 - 4*a^5*b^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a))/(b^11*x^3 + 3*a*b^10*x^2
+ 3*a^2*b^9*x + a^3*b^8)

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giac [B]  time = 1.32, size = 470, normalized size = 2.51 \begin {gather*} \frac {35 \, {\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \, {\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} + 21 \, {\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{8}} + \frac {3 \, b^{12} d^{7} x^{4} + 28 \, b^{12} c d^{6} x^{3} - 16 \, a b^{11} d^{7} x^{3} + 126 \, b^{12} c^{2} d^{5} x^{2} - 168 \, a b^{11} c d^{6} x^{2} + 60 \, a^{2} b^{10} d^{7} x^{2} + 420 \, b^{12} c^{3} d^{4} x - 1008 \, a b^{11} c^{2} d^{5} x + 840 \, a^{2} b^{10} c d^{6} x - 240 \, a^{3} b^{9} d^{7} x}{12 \, b^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="giac")

[Out]

35*(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7)*log(abs(b*x + a))/b^8 - 1/6*(
2*b^7*c^7 + 7*a*b^6*c^6*d + 42*a^2*b^5*c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^3*d^4 - 987*a^5*b^2*c^2*d
^5 + 518*a^6*b*c*d^6 - 107*a^7*d^7 + 126*(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 + 21*(b^7*c^6*d + 6*a*b^6*c^5*d^2 - 45*a^2*b^5*c^4*d^3 + 100*a^3*b^4*
c^3*d^4 - 105*a^4*b^3*c^2*d^5 + 54*a^5*b^2*c*d^6 - 11*a^6*b*d^7)*x)/((b*x + a)^3*b^8) + 1/12*(3*b^12*d^7*x^4 +
28*b^12*c*d^6*x^3 - 16*a*b^11*d^7*x^3 + 126*b^12*c^2*d^5*x^2 - 168*a*b^11*c*d^6*x^2 + 60*a^2*b^10*d^7*x^2 + 4
20*b^12*c^3*d^4*x - 1008*a*b^11*c^2*d^5*x + 840*a^2*b^10*c*d^6*x - 240*a^3*b^9*d^7*x)/b^16

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maple [B]  time = 0.01, size = 622, normalized size = 3.33 \begin {gather*} \frac {d^{7} x^{4}}{4 b^{4}}+\frac {a^{7} d^{7}}{3 \left (b x +a \right )^{3} b^{8}}-\frac {7 a^{6} c \,d^{6}}{3 \left (b x +a \right )^{3} b^{7}}+\frac {7 a^{5} c^{2} d^{5}}{\left (b x +a \right )^{3} b^{6}}-\frac {35 a^{4} c^{3} d^{4}}{3 \left (b x +a \right )^{3} b^{5}}+\frac {35 a^{3} c^{4} d^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {7 a^{2} c^{5} d^{2}}{\left (b x +a \right )^{3} b^{3}}+\frac {7 a \,c^{6} d}{3 \left (b x +a \right )^{3} b^{2}}-\frac {4 a \,d^{7} x^{3}}{3 b^{5}}-\frac {c^{7}}{3 \left (b x +a \right )^{3} b}+\frac {7 c \,d^{6} x^{3}}{3 b^{4}}-\frac {7 a^{6} d^{7}}{2 \left (b x +a \right )^{2} b^{8}}+\frac {21 a^{5} c \,d^{6}}{\left (b x +a \right )^{2} b^{7}}-\frac {105 a^{4} c^{2} d^{5}}{2 \left (b x +a \right )^{2} b^{6}}+\frac {70 a^{3} c^{3} d^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {105 a^{2} c^{4} d^{3}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {5 a^{2} d^{7} x^{2}}{b^{6}}+\frac {21 a \,c^{5} d^{2}}{\left (b x +a \right )^{2} b^{3}}-\frac {14 a c \,d^{6} x^{2}}{b^{5}}-\frac {7 c^{6} d}{2 \left (b x +a \right )^{2} b^{2}}+\frac {21 c^{2} d^{5} x^{2}}{2 b^{4}}+\frac {21 a^{5} d^{7}}{\left (b x +a \right ) b^{8}}-\frac {105 a^{4} c \,d^{6}}{\left (b x +a \right ) b^{7}}+\frac {35 a^{4} d^{7} \ln \left (b x +a \right )}{b^{8}}+\frac {210 a^{3} c^{2} d^{5}}{\left (b x +a \right ) b^{6}}-\frac {140 a^{3} c \,d^{6} \ln \left (b x +a \right )}{b^{7}}-\frac {20 a^{3} d^{7} x}{b^{7}}-\frac {210 a^{2} c^{3} d^{4}}{\left (b x +a \right ) b^{5}}+\frac {210 a^{2} c^{2} d^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {70 a^{2} c \,d^{6} x}{b^{6}}+\frac {105 a \,c^{4} d^{3}}{\left (b x +a \right ) b^{4}}-\frac {140 a \,c^{3} d^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {84 a \,c^{2} d^{5} x}{b^{5}}-\frac {21 c^{5} d^{2}}{\left (b x +a \right ) b^{3}}+\frac {35 c^{4} d^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {35 c^{3} d^{4} x}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a)^4,x)

[Out]

210/b^6*d^5/(b*x+a)*a^3*c^2-210/b^5*d^4/(b*x+a)*a^2*c^3+105/b^4*d^3/(b*x+a)*a*c^4-7/2/b^8*d^7/(b*x+a)^2*a^6-7/
2/b^2*d/(b*x+a)^2*c^6+35/b^8*d^7*ln(b*x+a)*a^4+35/b^4*d^3*ln(b*x+a)*c^4+21/b^8*d^7/(b*x+a)*a^5-21/b^3*d^2/(b*x
+a)*c^5-4/3*d^7/b^5*x^3*a+7/3*d^6/b^4*x^3*c+5*d^7/b^6*x^2*a^2+21/2*d^5/b^4*x^2*c^2-20*d^7/b^7*a^3*x+35*d^4/b^4
*c^3*x+1/3/b^8/(b*x+a)^3*a^7*d^7+1/4*d^7/b^4*x^4-1/3/b/(b*x+a)^3*c^7-84*d^5/b^5*a*c^2*x-7/3/b^7/(b*x+a)^3*a^6*
c*d^6+7/b^6/(b*x+a)^3*a^5*c^2*d^5-35/3/b^5/(b*x+a)^3*a^4*c^3*d^4+35/3/b^4/(b*x+a)^3*a^3*c^4*d^3-7/b^3/(b*x+a)^
3*a^2*c^5*d^2+7/3/b^2/(b*x+a)^3*a*c^6*d+21/b^7*d^6/(b*x+a)^2*a^5*c-105/2/b^6*d^5/(b*x+a)^2*a^4*c^2+70/b^5*d^4/
(b*x+a)^2*a^3*c^3-105/2/b^4*d^3/(b*x+a)^2*a^2*c^4+21/b^3*d^2/(b*x+a)^2*a*c^5-14*d^6/b^5*x^2*a*c+70*d^6/b^6*a^2
*c*x-140/b^7*d^6*ln(b*x+a)*a^3*c+210/b^6*d^5*ln(b*x+a)*a^2*c^2-140/b^5*d^4*ln(b*x+a)*a*c^3-105/b^7*d^6/(b*x+a)
*a^4*c

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maxima [B]  time = 1.63, size = 484, normalized size = 2.59 \begin {gather*} -\frac {2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \, {\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} + 21 \, {\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \, {\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} + \frac {3 \, b^{3} d^{7} x^{4} + 4 \, {\left (7 \, b^{3} c d^{6} - 4 \, a b^{2} d^{7}\right )} x^{3} + 6 \, {\left (21 \, b^{3} c^{2} d^{5} - 28 \, a b^{2} c d^{6} + 10 \, a^{2} b d^{7}\right )} x^{2} + 12 \, {\left (35 \, b^{3} c^{3} d^{4} - 84 \, a b^{2} c^{2} d^{5} + 70 \, a^{2} b c d^{6} - 20 \, a^{3} d^{7}\right )} x}{12 \, b^{7}} + \frac {35 \, {\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} 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)^4,x, algorithm="maxima")

[Out]

-1/6*(2*b^7*c^7 + 7*a*b^6*c^6*d + 42*a^2*b^5*c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^3*d^4 - 987*a^5*b^2
*c^2*d^5 + 518*a^6*b*c*d^6 - 107*a^7*d^7 + 126*(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 + 21*(b^7*c^6*d + 6*a*b^6*c^5*d^2 - 45*a^2*b^5*c^4*d^3 + 100*a^
3*b^4*c^3*d^4 - 105*a^4*b^3*c^2*d^5 + 54*a^5*b^2*c*d^6 - 11*a^6*b*d^7)*x)/(b^11*x^3 + 3*a*b^10*x^2 + 3*a^2*b^9
*x + a^3*b^8) + 1/12*(3*b^3*d^7*x^4 + 4*(7*b^3*c*d^6 - 4*a*b^2*d^7)*x^3 + 6*(21*b^3*c^2*d^5 - 28*a*b^2*c*d^6 +
10*a^2*b*d^7)*x^2 + 12*(35*b^3*c^3*d^4 - 84*a*b^2*c^2*d^5 + 70*a^2*b*c*d^6 - 20*a^3*d^7)*x)/b^7 + 35*(b^4*c^4
*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7)*log(b*x + a)/b^8

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mupad [B]  time = 0.29, size = 559, normalized size = 2.99 \begin {gather*} x^2\,\left (\frac {2\,a\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b}-\frac {3\,a^2\,d^7}{b^6}+\frac {21\,c^2\,d^5}{2\,b^4}\right )-x^3\,\left (\frac {4\,a\,d^7}{3\,b^5}-\frac {7\,c\,d^6}{3\,b^4}\right )-\frac {\frac {-107\,a^7\,d^7+518\,a^6\,b\,c\,d^6-987\,a^5\,b^2\,c^2\,d^5+910\,a^4\,b^3\,c^3\,d^4-385\,a^3\,b^4\,c^4\,d^3+42\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+2\,b^7\,c^7}{6\,b}+x\,\left (-\frac {77\,a^6\,d^7}{2}+189\,a^5\,b\,c\,d^6-\frac {735\,a^4\,b^2\,c^2\,d^5}{2}+350\,a^3\,b^3\,c^3\,d^4-\frac {315\,a^2\,b^4\,c^4\,d^3}{2}+21\,a\,b^5\,c^5\,d^2+\frac {7\,b^6\,c^6\,d}{2}\right )-x^2\,\left (21\,a^5\,b\,d^7-105\,a^4\,b^2\,c\,d^6+210\,a^3\,b^3\,c^2\,d^5-210\,a^2\,b^4\,c^3\,d^4+105\,a\,b^5\,c^4\,d^3-21\,b^6\,c^5\,d^2\right )}{a^3\,b^7+3\,a^2\,b^8\,x+3\,a\,b^9\,x^2+b^{10}\,x^3}-x\,\left (\frac {4\,a\,\left (\frac {4\,a\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b}-\frac {6\,a^2\,d^7}{b^6}+\frac {21\,c^2\,d^5}{b^4}\right )}{b}+\frac {4\,a^3\,d^7}{b^7}-\frac {35\,c^3\,d^4}{b^4}-\frac {6\,a^2\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (35\,a^4\,d^7-140\,a^3\,b\,c\,d^6+210\,a^2\,b^2\,c^2\,d^5-140\,a\,b^3\,c^3\,d^4+35\,b^4\,c^4\,d^3\right )}{b^8}+\frac {d^7\,x^4}{4\,b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x)^4,x)

[Out]

x^2*((2*a*((4*a*d^7)/b^5 - (7*c*d^6)/b^4))/b - (3*a^2*d^7)/b^6 + (21*c^2*d^5)/(2*b^4)) - x^3*((4*a*d^7)/(3*b^5
) - (7*c*d^6)/(3*b^4)) - ((2*b^7*c^7 - 107*a^7*d^7 + 42*a^2*b^5*c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^
3*d^4 - 987*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d + 518*a^6*b*c*d^6)/(6*b) + x*((7*b^6*c^6*d)/2 - (77*a^6*d^7)/2 + 2
1*a*b^5*c^5*d^2 - (315*a^2*b^4*c^4*d^3)/2 + 350*a^3*b^3*c^3*d^4 - (735*a^4*b^2*c^2*d^5)/2 + 189*a^5*b*c*d^6) -
x^2*(21*a^5*b*d^7 - 21*b^6*c^5*d^2 + 105*a*b^5*c^4*d^3 - 105*a^4*b^2*c*d^6 - 210*a^2*b^4*c^3*d^4 + 210*a^3*b^
3*c^2*d^5))/(a^3*b^7 + b^10*x^3 + 3*a^2*b^8*x + 3*a*b^9*x^2) - x*((4*a*((4*a*((4*a*d^7)/b^5 - (7*c*d^6)/b^4))/
b - (6*a^2*d^7)/b^6 + (21*c^2*d^5)/b^4))/b + (4*a^3*d^7)/b^7 - (35*c^3*d^4)/b^4 - (6*a^2*((4*a*d^7)/b^5 - (7*c
*d^6)/b^4))/b^2) + (log(a + b*x)*(35*a^4*d^7 + 35*b^4*c^4*d^3 - 140*a*b^3*c^3*d^4 + 210*a^2*b^2*c^2*d^5 - 140*
a^3*b*c*d^6))/b^8 + (d^7*x^4)/(4*b^4)

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sympy [B]  time = 6.12, size = 474, normalized size = 2.53 \begin {gather*} x^{3} \left (- \frac {4 a d^{7}}{3 b^{5}} + \frac {7 c d^{6}}{3 b^{4}}\right ) + x^{2} \left (\frac {5 a^{2} d^{7}}{b^{6}} - \frac {14 a c d^{6}}{b^{5}} + \frac {21 c^{2} d^{5}}{2 b^{4}}\right ) + x \left (- \frac {20 a^{3} d^{7}}{b^{7}} + \frac {70 a^{2} c d^{6}}{b^{6}} - \frac {84 a c^{2} d^{5}}{b^{5}} + \frac {35 c^{3} d^{4}}{b^{4}}\right ) + \frac {107 a^{7} d^{7} - 518 a^{6} b c d^{6} + 987 a^{5} b^{2} c^{2} d^{5} - 910 a^{4} b^{3} c^{3} d^{4} + 385 a^{3} b^{4} c^{4} d^{3} - 42 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 2 b^{7} c^{7} + x^{2} \left (126 a^{5} b^{2} d^{7} - 630 a^{4} b^{3} c d^{6} + 1260 a^{3} b^{4} c^{2} d^{5} - 1260 a^{2} b^{5} c^{3} d^{4} + 630 a b^{6} c^{4} d^{3} - 126 b^{7} c^{5} d^{2}\right ) + x \left (231 a^{6} b d^{7} - 1134 a^{5} b^{2} c d^{6} + 2205 a^{4} b^{3} c^{2} d^{5} - 2100 a^{3} b^{4} c^{3} d^{4} + 945 a^{2} b^{5} c^{4} d^{3} - 126 a b^{6} c^{5} d^{2} - 21 b^{7} c^{6} d\right )}{6 a^{3} b^{8} + 18 a^{2} b^{9} x + 18 a b^{10} x^{2} + 6 b^{11} x^{3}} + \frac {d^{7} x^{4}}{4 b^{4}} + \frac {35 d^{3} \left (a d - b c\right )^{4} \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)**4,x)

[Out]

x**3*(-4*a*d**7/(3*b**5) + 7*c*d**6/(3*b**4)) + x**2*(5*a**2*d**7/b**6 - 14*a*c*d**6/b**5 + 21*c**2*d**5/(2*b*
*4)) + x*(-20*a**3*d**7/b**7 + 70*a**2*c*d**6/b**6 - 84*a*c**2*d**5/b**5 + 35*c**3*d**4/b**4) + (107*a**7*d**7
- 518*a**6*b*c*d**6 + 987*a**5*b**2*c**2*d**5 - 910*a**4*b**3*c**3*d**4 + 385*a**3*b**4*c**4*d**3 - 42*a**2*b
**5*c**5*d**2 - 7*a*b**6*c**6*d - 2*b**7*c**7 + x**2*(126*a**5*b**2*d**7 - 630*a**4*b**3*c*d**6 + 1260*a**3*b*
*4*c**2*d**5 - 1260*a**2*b**5*c**3*d**4 + 630*a*b**6*c**4*d**3 - 126*b**7*c**5*d**2) + x*(231*a**6*b*d**7 - 11
34*a**5*b**2*c*d**6 + 2205*a**4*b**3*c**2*d**5 - 2100*a**3*b**4*c**3*d**4 + 945*a**2*b**5*c**4*d**3 - 126*a*b*
*6*c**5*d**2 - 21*b**7*c**6*d))/(6*a**3*b**8 + 18*a**2*b**9*x + 18*a*b**10*x**2 + 6*b**11*x**3) + d**7*x**4/(4
*b**4) + 35*d**3*(a*d - b*c)**4*log(a + b*x)/b**8

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