3.13.84 \(\int \frac {(c+d x)^7}{(a+b x)^2} \, dx\) [1284]

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

[Out]

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

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Rubi [A]
time = 0.16, 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 = {45} \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 {(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} \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 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)^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] Leaf count is larger than twice the leaf count of optimal. \(388\) vs. \(2(187)=374\).
time = 0.08, 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 )+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 )+420 d (b c-a d)^6 (a+b x) \log (a+b x)}{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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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(414\) vs. \(2(187)=374\).
time = 5.56, size = 404, normalized size = 2.16 \begin {gather*} \frac {7 d \text {Log}\left [a+b x\right ] \left (a+b x\right ) \left (a d-b c\right )^6+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 d^2 x \left (a+b x\right ) \left (6 a^5 d^5-35 a^4 b c d^4+84 a^3 b^2 c^2 d^3-105 a^2 b^3 c^3 d^2+70 a b^4 c^4 d-21 b^5 c^5\right )+\frac {b^2 d^3 x^2 \left (a+b x\right ) \left (5 a^4 d^4-28 a^3 b c d^3+63 a^2 b^2 c^2 d^2-70 a b^3 c^3 d+35 b^4 c^4\right )}{2}-\frac {b^3 d^4 x^3 \left (a+b x\right ) \left (4 a^3 d^3-21 a^2 b c d^2+42 a b^2 c^2 d-35 b^3 c^3\right )}{3}+\frac {b^4 d^5 x^4 \left (a+b x\right ) \left (3 a^2 d^2-14 a b c d+21 b^2 c^2\right )}{4}-\frac {b^5 d^6 x^5 \left (a+b x\right ) \left (2 a d-7 b c\right )}{5}+\frac {b^6 d^7 x^6 \left (a+b x\right )}{6}}{b^8 \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7 d Log[a + b x] (a + b x) (a d - b c) ^ 6 + a ^ 7 d ^ 7 - 7 a ^ 6 b c d ^ 6 + 21 a ^ 5 b ^ 2 c ^ 2 d ^ 5 - 3
5 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 d ^ 2 x (a + b x) (6 a ^ 5 d ^ 5 - 35 a ^ 4 b c d ^ 4 + 84 a ^ 3 b ^ 2 c ^ 2 d ^ 3 - 105 a ^ 2 b ^
3 c ^ 3 d ^ 2 + 70 a b ^ 4 c ^ 4 d - 21 b ^ 5 c ^ 5) + b ^ 2 d ^ 3 x ^ 2 (a + b x) (5 a ^ 4 d ^ 4 - 28 a ^ 3
b c d ^ 3 + 63 a ^ 2 b ^ 2 c ^ 2 d ^ 2 - 70 a b ^ 3 c ^ 3 d + 35 b ^ 4 c ^ 4) / 2 - b ^ 3 d ^ 4 x ^ 3 (a + b x
) (4 a ^ 3 d ^ 3 - 21 a ^ 2 b c d ^ 2 + 42 a b ^ 2 c ^ 2 d - 35 b ^ 3 c ^ 3) / 3 + b ^ 4 d ^ 5 x ^ 4 (a + b x)
(3 a ^ 2 d ^ 2 - 14 a b c d + 21 b ^ 2 c ^ 2) / 4 - b ^ 5 d ^ 6 x ^ 5 (a + b x) (2 a d - 7 b c) / 5 + b ^ 6 d
^ 7 x ^ 6 (a + b x) / 6) / (b ^ 8 (a + b x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs. \(2(177)=354\).
time = 0.17, size = 479, normalized size = 2.56

method	result	size
norman	\(\frac {\frac {d^{7} x^{7}}{6 b}-\frac {7 d^{2} \left (a^{5} d^{5}-6 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}+15 a \,b^{4} c^{4} d -6 b^{5} c^{5}\right ) x^{2}}{2 b^{6}}+\frac {7 d^{3} \left (a^{4} d^{4}-6 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}-20 a \,b^{3} c^{3} d +15 b^{4} c^{4}\right ) x^{3}}{6 b^{5}}-\frac {7 d^{4} \left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -20 b^{3} c^{3}\right ) x^{4}}{12 b^{4}}+\frac {7 d^{5} \left (a^{2} d^{2}-6 a b c d +15 b^{2} c^{2}\right ) x^{5}}{20 b^{3}}-\frac {7 d^{6} \left (a d -6 b c \right ) x^{6}}{30 b^{2}}-\frac {\left (7 a^{7} d^{7}-42 a^{6} b c \,d^{6}+105 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right ) x}{b^{7} a}}{b x +a}+\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(448\)
default	\(-\frac {d^{2} \left (-\frac {1}{6} d^{5} x^{6} b^{5}+\frac {2}{5} a \,b^{4} d^{5} x^{5}-\frac {7}{5} b^{5} c \,d^{4} x^{5}-\frac {3}{4} a^{2} b^{3} d^{5} x^{4}+\frac {7}{2} a \,b^{4} c \,d^{4} x^{4}-\frac {21}{4} b^{5} c^{2} d^{3} x^{4}+\frac {4}{3} a^{3} b^{2} d^{5} x^{3}-7 a^{2} b^{3} c \,d^{4} x^{3}+14 a \,b^{4} c^{2} d^{3} x^{3}-\frac {35}{3} b^{5} c^{3} d^{2} x^{3}-\frac {5}{2} a^{4} b \,d^{5} x^{2}+14 a^{3} b^{2} c \,d^{4} x^{2}-\frac {63}{2} a^{2} b^{3} c^{2} d^{3} x^{2}+35 a \,b^{4} c^{3} d^{2} x^{2}-\frac {35}{2} b^{5} c^{4} d \,x^{2}+6 a^{5} d^{5} x -35 a^{4} b c \,d^{4} x +84 a^{3} b^{2} c^{2} d^{3} x -105 a^{2} b^{3} c^{3} d^{2} x +70 a \,b^{4} c^{4} d x -21 b^{5} c^{5} x \right )}{b^{7}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{b^{8} \left (b x +a \right )}+\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(479\)
risch	\(\frac {d^{7} x^{6}}{6 b^{2}}-\frac {c^{7}}{b \left (b x +a \right )}-\frac {42 d^{6} \ln \left (b x +a \right ) a^{5} c}{b^{7}}+\frac {105 d^{5} \ln \left (b x +a \right ) a^{4} c^{2}}{b^{6}}-\frac {140 d^{4} \ln \left (b x +a \right ) a^{3} c^{3}}{b^{5}}+\frac {105 d^{3} \ln \left (b x +a \right ) a^{2} c^{4}}{b^{4}}-\frac {42 d^{2} \ln \left (b x +a \right ) a \,c^{5}}{b^{3}}-\frac {2 d^{7} a \,x^{5}}{5 b^{3}}+\frac {7 d^{6} c \,x^{5}}{5 b^{2}}+\frac {3 d^{7} a^{2} x^{4}}{4 b^{4}}+\frac {21 d^{5} c^{2} x^{4}}{4 b^{2}}-\frac {4 d^{7} a^{3} x^{3}}{3 b^{5}}+\frac {35 d^{4} c^{3} x^{3}}{3 b^{2}}+\frac {5 d^{7} a^{4} x^{2}}{2 b^{6}}+\frac {35 d^{3} c^{4} x^{2}}{2 b^{2}}-\frac {6 d^{7} a^{5} x}{b^{7}}+\frac {21 d^{2} c^{5} x}{b^{2}}+\frac {a^{7} d^{7}}{b^{8} \left (b x +a \right )}+\frac {7 d^{7} \ln \left (b x +a \right ) a^{6}}{b^{8}}+\frac {7 d \ln \left (b x +a \right ) c^{6}}{b^{2}}-\frac {14 d^{5} a \,c^{2} x^{3}}{b^{3}}-\frac {14 d^{6} a^{3} c \,x^{2}}{b^{5}}+\frac {63 d^{5} a^{2} c^{2} x^{2}}{2 b^{4}}-\frac {35 d^{4} a \,c^{3} x^{2}}{b^{3}}+\frac {35 d^{6} a^{4} c x}{b^{6}}-\frac {84 d^{5} a^{3} c^{2} x}{b^{5}}+\frac {105 d^{4} a^{2} c^{3} x}{b^{4}}-\frac {70 d^{3} a \,c^{4} x}{b^{3}}-\frac {7 a^{6} c \,d^{6}}{b^{7} \left (b x +a \right )}+\frac {21 a^{5} c^{2} d^{5}}{b^{6} \left (b x +a \right )}-\frac {35 a^{4} c^{3} d^{4}}{b^{5} \left (b x +a \right )}+\frac {35 a^{3} c^{4} d^{3}}{b^{4} \left (b x +a \right )}-\frac {21 a^{2} c^{5} d^{2}}{b^{3} \left (b x +a \right )}+\frac {7 a \,c^{6} d}{b^{2} \left (b x +a \right )}-\frac {7 d^{6} a c \,x^{4}}{2 b^{3}}+\frac {7 d^{6} a^{2} c \,x^{3}}{b^{4}}\)	\(571\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^2/b^7*(-1/6*d^5*x^6*b^5+2/5*a*b^4*d^5*x^5-7/5*b^5*c*d^4*x^5-3/4*a^2*b^3*d^5*x^4+7/2*a*b^4*c*d^4*x^4-21/4*b^
5*c^2*d^3*x^4+4/3*a^3*b^2*d^5*x^3-7*a^2*b^3*c*d^4*x^3+14*a*b^4*c^2*d^3*x^3-35/3*b^5*c^3*d^2*x^3-5/2*a^4*b*d^5*
x^2+14*a^3*b^2*c*d^4*x^2-63/2*a^2*b^3*c^2*d^3*x^2+35*a*b^4*c^3*d^2*x^2-35/2*b^5*c^4*d*x^2+6*a^5*d^5*x-35*a^4*b
*c*d^4*x+84*a^3*b^2*c^2*d^3*x-105*a^2*b^3*c^3*d^2*x+70*a*b^4*c^4*d*x-21*b^5*c^5*x)-1/b^8*(-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*x+a)+
7/b^8*d*(a^6*d^6-6*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)
*ln(b*x+a)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (177) = 354\).
time = 0.28, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (177) = 354\).
time = 0.29, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (172) = 344\).
time = 0.85, 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} \cdot \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} \cdot \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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (177) = 354\).
time = 0.00, size = 530, normalized size = 2.83 \begin {gather*} \frac {\frac {1}{6} x^{6} d^{7} b^{10}-\frac {2}{5} x^{5} d^{7} b^{9} a+\frac {7}{5} x^{5} d^{6} c b^{10}+\frac {3}{4} x^{4} d^{7} b^{8} a^{2}-\frac {7}{2} x^{4} d^{6} c b^{9} a+\frac {21}{4} x^{4} d^{5} c^{2} b^{10}-\frac {4}{3} x^{3} d^{7} b^{7} a^{3}+7 x^{3} d^{6} c b^{8} a^{2}-14 x^{3} d^{5} c^{2} b^{9} a+\frac {35}{3} x^{3} d^{4} c^{3} b^{10}+\frac {5}{2} x^{2} d^{7} b^{6} a^{4}-14 x^{2} d^{6} c b^{7} a^{3}+\frac {63}{2} x^{2} d^{5} c^{2} b^{8} a^{2}-35 x^{2} d^{4} c^{3} b^{9} a+\frac {35}{2} x^{2} d^{3} c^{4} b^{10}-6 x d^{7} b^{5} a^{5}+35 x d^{6} c b^{6} a^{4}-84 x d^{5} c^{2} b^{7} a^{3}+105 x d^{4} c^{3} b^{8} a^{2}-70 x d^{3} c^{4} b^{9} a+21 x d^{2} c^{5} b^{10}}{b^{12}}+\frac {d^{7} a^{7}-7 d^{6} b a^{6} c+21 d^{5} b^{2} a^{5} c^{2}-35 d^{4} b^{3} a^{4} c^{3}+35 d^{3} b^{4} a^{3} c^{4}-21 d^{2} b^{5} a^{2} c^{5}+7 d b^{6} a c^{6}-b^{7} c^{7}}{b^{8} \left (x b+a\right )}+\frac {\left (7 d^{7} a^{6}-42 d^{6} c b a^{5}+105 d^{5} c^{2} b^{2} a^{4}-140 d^{4} c^{3} b^{3} a^{3}+105 d^{3} c^{4} b^{4} a^{2}-42 d^{2} c^{5} b^{5} a+7 d c^{6} b^{6}\right ) \ln \left |x b+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)

[Out]

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^8 - (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*x + a)*b^8) + 1/60*(10*b^10*d^7*x^6 + 84*b^10
*c*d^6*x^5 - 24*a*b^9*d^7*x^5 + 315*b^10*c^2*d^5*x^4 - 210*a*b^9*c*d^6*x^4 + 45*a^2*b^8*d^7*x^4 + 700*b^10*c^3
*d^4*x^3 - 840*a*b^9*c^2*d^5*x^3 + 420*a^2*b^8*c*d^6*x^3 - 80*a^3*b^7*d^7*x^3 + 1050*b^10*c^4*d^3*x^2 - 2100*a
*b^9*c^3*d^4*x^2 + 1890*a^2*b^8*c^2*d^5*x^2 - 840*a^3*b^7*c*d^6*x^2 + 150*a^4*b^6*d^7*x^2 + 1260*b^10*c^5*d^2*
x - 4200*a*b^9*c^4*d^3*x + 6300*a^2*b^8*c^3*d^4*x - 5040*a^3*b^7*c^2*d^5*x + 2100*a^4*b^6*c*d^6*x - 360*a^5*b^
5*d^7*x)/b^12

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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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