3.13.83 \(\int \frac {(c+d x)^7}{a+b x} \, dx\) [1283]

Optimal. Leaf size=169 \[ \frac {d (b c-a d)^6 x}{b^7}+\frac {(b c-a d)^5 (c+d x)^2}{2 b^6}+\frac {(b c-a d)^4 (c+d x)^3}{3 b^5}+\frac {(b c-a d)^3 (c+d x)^4}{4 b^4}+\frac {(b c-a d)^2 (c+d x)^5}{5 b^3}+\frac {(b c-a d) (c+d x)^6}{6 b^2}+\frac {(c+d x)^7}{7 b}+\frac {(b c-a d)^7 \log (a+b x)}{b^8} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 169, 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 = {45} \begin {gather*} \frac {(b c-a d)^7 \log (a+b x)}{b^8}+\frac {d x (b c-a d)^6}{b^7}+\frac {(c+d x)^2 (b c-a d)^5}{2 b^6}+\frac {(c+d x)^3 (b c-a d)^4}{3 b^5}+\frac {(c+d x)^4 (b c-a d)^3}{4 b^4}+\frac {(c+d x)^5 (b c-a d)^2}{5 b^3}+\frac {(c+d x)^6 (b c-a d)}{6 b^2}+\frac {(c+d x)^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^7/(a + b*x),x]

[Out]

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

Rule 45

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

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Mathematica [A]
time = 0.10, size = 304, normalized size = 1.80 \begin {gather*} \frac {d x \left (420 a^6 d^6-210 a^5 b d^5 (14 c+d x)+70 a^4 b^2 d^4 \left (126 c^2+21 c d x+2 d^2 x^2\right )-35 a^3 b^3 d^3 \left (420 c^3+126 c^2 d x+28 c d^2 x^2+3 d^3 x^3\right )+21 a^2 b^4 d^2 \left (700 c^4+350 c^3 d x+140 c^2 d^2 x^2+35 c d^3 x^3+4 d^4 x^4\right )-7 a b^5 d \left (1260 c^5+1050 c^4 d x+700 c^3 d^2 x^2+315 c^2 d^3 x^3+84 c d^4 x^4+10 d^5 x^5\right )+b^6 \left (2940 c^6+4410 c^5 d x+4900 c^4 d^2 x^2+3675 c^3 d^3 x^3+1764 c^2 d^4 x^4+490 c d^5 x^5+60 d^6 x^6\right )\right )}{420 b^7}+\frac {(b c-a d)^7 \log (a+b x)}{b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^7/(a + b*x),x]

[Out]

(d*x*(420*a^6*d^6 - 210*a^5*b*d^5*(14*c + d*x) + 70*a^4*b^2*d^4*(126*c^2 + 21*c*d*x + 2*d^2*x^2) - 35*a^3*b^3*
d^3*(420*c^3 + 126*c^2*d*x + 28*c*d^2*x^2 + 3*d^3*x^3) + 21*a^2*b^4*d^2*(700*c^4 + 350*c^3*d*x + 140*c^2*d^2*x
^2 + 35*c*d^3*x^3 + 4*d^4*x^4) - 7*a*b^5*d*(1260*c^5 + 1050*c^4*d*x + 700*c^3*d^2*x^2 + 315*c^2*d^3*x^3 + 84*c
*d^4*x^4 + 10*d^5*x^5) + b^6*(2940*c^6 + 4410*c^5*d*x + 4900*c^4*d^2*x^2 + 3675*c^3*d^3*x^3 + 1764*c^2*d^4*x^4
+ 490*c*d^5*x^5 + 60*d^6*x^6)))/(420*b^7) + ((b*c - a*d)^7*Log[a + b*x])/b^8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(157)=314\).
time = 0.14, size = 491, normalized size = 2.91

method	result	size
norman	\(\frac {d \left (a^{6} d^{6}-7 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}-35 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}-21 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x}{b^{7}}+\frac {d^{7} x^{7}}{7 b}-\frac {d^{2} \left (a^{5} d^{5}-7 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}-35 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d -21 b^{5} c^{5}\right ) x^{2}}{2 b^{6}}+\frac {d^{3} \left (a^{4} d^{4}-7 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}-35 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) x^{3}}{3 b^{5}}-\frac {d^{4} \left (a^{3} d^{3}-7 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d -35 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}+\frac {d^{5} \left (a^{2} d^{2}-7 a b c d +21 b^{2} c^{2}\right ) x^{5}}{5 b^{3}}-\frac {d^{6} \left (a d -7 b c \right ) x^{6}}{6 b^{2}}-\frac {\left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(435\)
default	\(\frac {d \left (-\frac {1}{6} a \,b^{5} d^{6} x^{6}+\frac {7}{6} b^{6} c \,d^{5} x^{6}+\frac {1}{5} a^{2} b^{4} d^{6} x^{5}+\frac {21}{5} b^{6} c^{2} d^{4} x^{5}-\frac {1}{4} a^{3} b^{3} d^{6} x^{4}+\frac {35}{4} b^{6} c^{3} d^{3} x^{4}+\frac {1}{3} a^{4} b^{2} d^{6} x^{3}+\frac {35}{3} b^{6} c^{4} d^{2} x^{3}-\frac {1}{2} a^{5} b \,d^{6} x^{2}+\frac {1}{7} d^{6} x^{7} b^{6}+a^{6} d^{6} x +7 b^{6} c^{6} x -\frac {21}{4} a \,b^{5} c^{2} d^{4} x^{4}-\frac {7}{3} a^{3} b^{3} c \,d^{5} x^{3}+7 a^{2} b^{4} c^{2} d^{4} x^{3}-\frac {35}{3} a \,b^{5} c^{3} d^{3} x^{3}+\frac {7}{2} a^{4} b^{2} c \,d^{5} x^{2}-\frac {21}{2} a^{3} b^{3} c^{2} d^{4} x^{2}+\frac {35}{2} a^{2} b^{4} c^{3} d^{3} x^{2}-\frac {35}{2} a \,b^{5} c^{4} d^{2} x^{2}-7 a^{5} b c \,d^{5} x +21 a^{4} b^{2} c^{2} d^{4} x -35 a^{3} b^{3} c^{3} d^{3} x +35 a^{2} b^{4} c^{4} d^{2} x -21 a \,b^{5} c^{5} d x +\frac {21}{2} b^{6} c^{5} d \,x^{2}-\frac {7}{5} a \,b^{5} c \,d^{5} x^{5}+\frac {7}{4} a^{2} b^{4} c \,d^{5} x^{4}\right )}{b^{7}}+\frac {\left (-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(491\)
risch	\(\frac {\ln \left (b x +a \right ) c^{7}}{b}-\frac {d^{7} a \,x^{6}}{6 b^{2}}+\frac {7 d^{6} c \,x^{6}}{6 b}+\frac {d^{7} a^{2} x^{5}}{5 b^{3}}+\frac {21 d^{5} c^{2} x^{5}}{5 b}-\frac {d^{7} a^{3} x^{4}}{4 b^{4}}+\frac {35 d^{4} c^{3} x^{4}}{4 b}+\frac {d^{7} a^{4} x^{3}}{3 b^{5}}+\frac {35 d^{3} c^{4} x^{3}}{3 b}-\frac {d^{7} a^{5} x^{2}}{2 b^{6}}+\frac {d^{7} a^{6} x}{b^{7}}+\frac {7 d \,c^{6} x}{b}+\frac {21 d^{2} c^{5} x^{2}}{2 b}-\frac {\ln \left (b x +a \right ) a^{7} d^{7}}{b^{8}}+\frac {21 \ln \left (b x +a \right ) a^{2} c^{5} d^{2}}{b^{3}}-\frac {7 \ln \left (b x +a \right ) a \,c^{6} d}{b^{2}}-\frac {7 d^{6} a c \,x^{5}}{5 b^{2}}+\frac {7 d^{6} a^{2} c \,x^{4}}{4 b^{3}}+\frac {7 \ln \left (b x +a \right ) a^{6} c \,d^{6}}{b^{7}}-\frac {21 \ln \left (b x +a \right ) a^{5} c^{2} d^{5}}{b^{6}}+\frac {35 \ln \left (b x +a \right ) a^{4} c^{3} d^{4}}{b^{5}}-\frac {35 \ln \left (b x +a \right ) a^{3} c^{4} d^{3}}{b^{4}}+\frac {7 d^{5} a^{2} c^{2} x^{3}}{b^{3}}-\frac {35 d^{4} a \,c^{3} x^{3}}{3 b^{2}}+\frac {7 d^{6} a^{4} c \,x^{2}}{2 b^{5}}-\frac {21 d^{5} a^{3} c^{2} x^{2}}{2 b^{4}}+\frac {35 d^{4} a^{2} c^{3} x^{2}}{2 b^{3}}-\frac {35 d^{3} a \,c^{4} x^{2}}{2 b^{2}}-\frac {7 d^{6} a^{5} c x}{b^{6}}+\frac {21 d^{5} a^{4} c^{2} x}{b^{5}}-\frac {35 d^{4} a^{3} c^{3} x}{b^{4}}+\frac {35 d^{3} a^{2} c^{4} x}{b^{3}}-\frac {21 d^{2} a \,c^{5} x}{b^{2}}-\frac {7 d^{6} a^{3} c \,x^{3}}{3 b^{4}}-\frac {21 d^{5} a \,c^{2} x^{4}}{4 b^{2}}+\frac {d^{7} x^{7}}{7 b}\)	\(539\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^7/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (157) = 314\).
time = 0.28, size = 460, normalized size = 2.72 \begin {gather*} \frac {60 \, b^{6} d^{7} x^{7} + 70 \, {\left (7 \, b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 84 \, {\left (21 \, b^{6} c^{2} d^{5} - 7 \, a b^{5} c d^{6} + a^{2} b^{4} d^{7}\right )} x^{5} + 105 \, {\left (35 \, b^{6} c^{3} d^{4} - 21 \, a b^{5} c^{2} d^{5} + 7 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )} x^{4} + 140 \, {\left (35 \, b^{6} c^{4} d^{3} - 35 \, a b^{5} c^{3} d^{4} + 21 \, a^{2} b^{4} c^{2} d^{5} - 7 \, a^{3} b^{3} c d^{6} + a^{4} b^{2} d^{7}\right )} x^{3} + 210 \, {\left (21 \, b^{6} c^{5} d^{2} - 35 \, a b^{5} c^{4} d^{3} + 35 \, a^{2} b^{4} c^{3} d^{4} - 21 \, a^{3} b^{3} c^{2} d^{5} + 7 \, a^{4} b^{2} c d^{6} - a^{5} b d^{7}\right )} x^{2} + 420 \, {\left (7 \, b^{6} c^{6} d - 21 \, a b^{5} c^{5} d^{2} + 35 \, a^{2} b^{4} c^{4} d^{3} - 35 \, a^{3} b^{3} c^{3} d^{4} + 21 \, a^{4} b^{2} c^{2} d^{5} - 7 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x}{420 \, b^{7}} + \frac {{\left (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}\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),x, algorithm="maxima")

[Out]

1/420*(60*b^6*d^7*x^7 + 70*(7*b^6*c*d^6 - a*b^5*d^7)*x^6 + 84*(21*b^6*c^2*d^5 - 7*a*b^5*c*d^6 + a^2*b^4*d^7)*x
^5 + 105*(35*b^6*c^3*d^4 - 21*a*b^5*c^2*d^5 + 7*a^2*b^4*c*d^6 - a^3*b^3*d^7)*x^4 + 140*(35*b^6*c^4*d^3 - 35*a*
b^5*c^3*d^4 + 21*a^2*b^4*c^2*d^5 - 7*a^3*b^3*c*d^6 + a^4*b^2*d^7)*x^3 + 210*(21*b^6*c^5*d^2 - 35*a*b^5*c^4*d^3
+ 35*a^2*b^4*c^3*d^4 - 21*a^3*b^3*c^2*d^5 + 7*a^4*b^2*c*d^6 - a^5*b*d^7)*x^2 + 420*(7*b^6*c^6*d - 21*a*b^5*c^
5*d^2 + 35*a^2*b^4*c^4*d^3 - 35*a^3*b^3*c^3*d^4 + 21*a^4*b^2*c^2*d^5 - 7*a^5*b*c*d^6 + a^6*d^7)*x)/b^7 + (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)*log(b*x + a)/b^8

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (157) = 314\).
time = 1.09, size = 462, normalized size = 2.73 \begin {gather*} \frac {60 \, b^{7} d^{7} x^{7} + 70 \, {\left (7 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \, {\left (21 \, b^{7} c^{2} d^{5} - 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \, {\left (35 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 140 \, {\left (35 \, b^{7} c^{4} d^{3} - 35 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} - 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 210 \, {\left (21 \, b^{7} c^{5} d^{2} - 35 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} - 21 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 420 \, {\left (7 \, b^{7} c^{6} d - 21 \, a b^{6} c^{5} d^{2} + 35 \, 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} - 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x + 420 \, {\left (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}\right )} \log \left (b x + a\right )}{420 \, b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(60*b^7*d^7*x^7 + 70*(7*b^7*c*d^6 - a*b^6*d^7)*x^6 + 84*(21*b^7*c^2*d^5 - 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x
^5 + 105*(35*b^7*c^3*d^4 - 21*a*b^6*c^2*d^5 + 7*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 140*(35*b^7*c^4*d^3 - 35*a*
b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 - 7*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 210*(21*b^7*c^5*d^2 - 35*a*b^6*c^4*d^3
+ 35*a^2*b^5*c^3*d^4 - 21*a^3*b^4*c^2*d^5 + 7*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 420*(7*b^7*c^6*d - 21*a*b^6*
c^5*d^2 + 35*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 - 7*a^5*b^2*c*d^6 + a^6*b*d^7)*x + 420*
(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)*log(b*x + a))/b^8

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (144) = 288\).
time = 0.48, size = 408, normalized size = 2.41 \begin {gather*} x^{6} \left (- \frac {a d^{7}}{6 b^{2}} + \frac {7 c d^{6}}{6 b}\right ) + x^{5} \left (\frac {a^{2} d^{7}}{5 b^{3}} - \frac {7 a c d^{6}}{5 b^{2}} + \frac {21 c^{2} d^{5}}{5 b}\right ) + x^{4} \left (- \frac {a^{3} d^{7}}{4 b^{4}} + \frac {7 a^{2} c d^{6}}{4 b^{3}} - \frac {21 a c^{2} d^{5}}{4 b^{2}} + \frac {35 c^{3} d^{4}}{4 b}\right ) + x^{3} \left (\frac {a^{4} d^{7}}{3 b^{5}} - \frac {7 a^{3} c d^{6}}{3 b^{4}} + \frac {7 a^{2} c^{2} d^{5}}{b^{3}} - \frac {35 a c^{3} d^{4}}{3 b^{2}} + \frac {35 c^{4} d^{3}}{3 b}\right ) + x^{2} \left (- \frac {a^{5} d^{7}}{2 b^{6}} + \frac {7 a^{4} c d^{6}}{2 b^{5}} - \frac {21 a^{3} c^{2} d^{5}}{2 b^{4}} + \frac {35 a^{2} c^{3} d^{4}}{2 b^{3}} - \frac {35 a c^{4} d^{3}}{2 b^{2}} + \frac {21 c^{5} d^{2}}{2 b}\right ) + x \left (\frac {a^{6} d^{7}}{b^{7}} - \frac {7 a^{5} c d^{6}}{b^{6}} + \frac {21 a^{4} c^{2} d^{5}}{b^{5}} - \frac {35 a^{3} c^{3} d^{4}}{b^{4}} + \frac {35 a^{2} c^{4} d^{3}}{b^{3}} - \frac {21 a c^{5} d^{2}}{b^{2}} + \frac {7 c^{6} d}{b}\right ) + \frac {d^{7} x^{7}}{7 b} - \frac {\left (a d - b c\right )^{7} \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),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (157) = 314\).
time = 0.75, size = 497, normalized size = 2.94 \begin {gather*} \frac {60 \, b^{6} d^{7} x^{7} + 490 \, b^{6} c d^{6} x^{6} - 70 \, a b^{5} d^{7} x^{6} + 1764 \, b^{6} c^{2} d^{5} x^{5} - 588 \, a b^{5} c d^{6} x^{5} + 84 \, a^{2} b^{4} d^{7} x^{5} + 3675 \, b^{6} c^{3} d^{4} x^{4} - 2205 \, a b^{5} c^{2} d^{5} x^{4} + 735 \, a^{2} b^{4} c d^{6} x^{4} - 105 \, a^{3} b^{3} d^{7} x^{4} + 4900 \, b^{6} c^{4} d^{3} x^{3} - 4900 \, a b^{5} c^{3} d^{4} x^{3} + 2940 \, a^{2} b^{4} c^{2} d^{5} x^{3} - 980 \, a^{3} b^{3} c d^{6} x^{3} + 140 \, a^{4} b^{2} d^{7} x^{3} + 4410 \, b^{6} c^{5} d^{2} x^{2} - 7350 \, a b^{5} c^{4} d^{3} x^{2} + 7350 \, a^{2} b^{4} c^{3} d^{4} x^{2} - 4410 \, a^{3} b^{3} c^{2} d^{5} x^{2} + 1470 \, a^{4} b^{2} c d^{6} x^{2} - 210 \, a^{5} b d^{7} x^{2} + 2940 \, b^{6} c^{6} d x - 8820 \, a b^{5} c^{5} d^{2} x + 14700 \, a^{2} b^{4} c^{4} d^{3} x - 14700 \, a^{3} b^{3} c^{3} d^{4} x + 8820 \, a^{4} b^{2} c^{2} d^{5} x - 2940 \, a^{5} b c d^{6} x + 420 \, a^{6} d^{7} x}{420 \, b^{7}} + \frac {{\left (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}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(60*b^6*d^7*x^7 + 490*b^6*c*d^6*x^6 - 70*a*b^5*d^7*x^6 + 1764*b^6*c^2*d^5*x^5 - 588*a*b^5*c*d^6*x^5 + 84
*a^2*b^4*d^7*x^5 + 3675*b^6*c^3*d^4*x^4 - 2205*a*b^5*c^2*d^5*x^4 + 735*a^2*b^4*c*d^6*x^4 - 105*a^3*b^3*d^7*x^4
+ 4900*b^6*c^4*d^3*x^3 - 4900*a*b^5*c^3*d^4*x^3 + 2940*a^2*b^4*c^2*d^5*x^3 - 980*a^3*b^3*c*d^6*x^3 + 140*a^4*
b^2*d^7*x^3 + 4410*b^6*c^5*d^2*x^2 - 7350*a*b^5*c^4*d^3*x^2 + 7350*a^2*b^4*c^3*d^4*x^2 - 4410*a^3*b^3*c^2*d^5*
x^2 + 1470*a^4*b^2*c*d^6*x^2 - 210*a^5*b*d^7*x^2 + 2940*b^6*c^6*d*x - 8820*a*b^5*c^5*d^2*x + 14700*a^2*b^4*c^4
*d^3*x - 14700*a^3*b^3*c^3*d^4*x + 8820*a^4*b^2*c^2*d^5*x - 2940*a^5*b*c*d^6*x + 420*a^6*d^7*x)/b^7 + (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)*log(abs(b*x + a))/b^8

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Mupad [B]
time = 0.22, size = 509, normalized size = 3.01 \begin {gather*} x\,\left (\frac {7\,c^6\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{b}-\frac {35\,c^4\,d^3}{b}\right )}{b}+\frac {21\,c^5\,d^2}{b}\right )}{b}\right )-x^6\,\left (\frac {a\,d^7}{6\,b^2}-\frac {7\,c\,d^6}{6\,b}\right )+x^4\,\left (\frac {35\,c^3\,d^4}{4\,b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{4\,b}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{b}-\frac {35\,c^4\,d^3}{b}\right )}{2\,b}+\frac {21\,c^5\,d^2}{2\,b}\right )+x^5\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{5\,b}+\frac {21\,c^2\,d^5}{5\,b}\right )-x^3\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{3\,b}-\frac {35\,c^4\,d^3}{3\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (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\right )}{b^8}+\frac {d^7\,x^7}{7\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^7/(a + b*x),x)

[Out]

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

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