3.13.86 \(\int \frac {(c+d x)^7}{(a+b x)^4} \, dx\) [1286]

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

[Out]

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

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Rubi [A]
time = 0.15, 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)^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} \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 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)^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 = 0.07, size = 199, normalized size = 1.06 \begin {gather*} \frac {12 b d^4 \left (35 b^3 c^3-84 a b^2 c^2 d+70 a^2 b c d^2-20 a^3 d^3\right ) x+6 b^2 d^5 \left (21 b^2 c^2-28 a b c d+10 a^2 d^2\right ) x^2+4 b^3 d^6 (7 b c-4 a d) x^3+3 b^4 d^7 x^4-\frac {4 (b c-a d)^7}{(a+b x)^3}-\frac {42 d (b c-a d)^6}{(a+b x)^2}+\frac {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} \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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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(552\) vs. \(2(187)=374\).
time = 9.09, size = 550, normalized size = 2.94 \begin {gather*} \frac {214 a^7 d^7-1036 a^6 b c d^6+1974 a^5 b^2 c^2 d^5-1820 a^4 b^3 c^3 d^4+770 a^3 b^4 c^4 d^3-84 a^2 b^5 c^5 d^2-14 a b^6 c^6 d-4 b^7 c^7-42 b d x \left (-11 a^6 d^6+54 a^5 b c d^5-105 a^4 b^2 c^2 d^4+100 a^3 b^3 c^3 d^3-45 a^2 b^4 c^4 d^2+6 a b^5 c^5 d+b^6 c^6\right )+252 b^2 d^2 x^2 \left (a^5 d^5-5 a^4 b c d^4+10 a^3 b^2 c^2 d^3-10 a^2 b^3 c^3 d^2+5 a b^4 c^4 d-b^5 c^5\right )+420 d^3 \text {Log}\left [a+b x\right ] \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \left (a d-b c\right )^4-12 b d^4 x \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \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 )+6 b^2 d^5 x^2 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \left (10 a^2 d^2-28 a b c d+21 b^2 c^2\right )-4 b^3 d^6 x^3 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right ) \left (4 a d-7 b c\right )+3 b^4 d^7 x^4 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )}{12 b^8 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(214 a ^ 7 d ^ 7 - 1036 a ^ 6 b c d ^ 6 + 1974 a ^ 5 b ^ 2 c ^ 2 d ^ 5 - 1820 a ^ 4 b ^ 3 c ^ 3 d ^ 4 + 770 a
^ 3 b ^ 4 c ^ 4 d ^ 3 - 84 a ^ 2 b ^ 5 c ^ 5 d ^ 2 - 14 a b ^ 6 c ^ 6 d - 4 b ^ 7 c ^ 7 - 42 b d x (-11 a ^ 6
d ^ 6 + 54 a ^ 5 b c d ^ 5 - 105 a ^ 4 b ^ 2 c ^ 2 d ^ 4 + 100 a ^ 3 b ^ 3 c ^ 3 d ^ 3 - 45 a ^ 2 b ^ 4 c ^ 4
d ^ 2 + 6 a b ^ 5 c ^ 5 d + b ^ 6 c ^ 6) + 252 b ^ 2 d ^ 2 x ^ 2 (a ^ 5 d ^ 5 - 5 a ^ 4 b c d ^ 4 + 10 a ^ 3 b
^ 2 c ^ 2 d ^ 3 - 10 a ^ 2 b ^ 3 c ^ 3 d ^ 2 + 5 a b ^ 4 c ^ 4 d - b ^ 5 c ^ 5) + 420 d ^ 3 Log[a + b x] (a ^
3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3) (a d - b c) ^ 4 - 12 b d ^ 4 x (a ^ 3 + 3 a ^ 2 b x + 3 a b
^ 2 x ^ 2 + b ^ 3 x ^ 3) (20 a ^ 3 d ^ 3 - 70 a ^ 2 b c d ^ 2 + 84 a b ^ 2 c ^ 2 d - 35 b ^ 3 c ^ 3) + 6 b ^ 2
d ^ 5 x ^ 2 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3) (10 a ^ 2 d ^ 2 - 28 a b c d + 21 b ^ 2 c ^
2) - 4 b ^ 3 d ^ 6 x ^ 3 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3) (4 a d - 7 b c) + 3 b ^ 4 d ^
7 x ^ 4 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3)) / (12 b ^ 8 (a ^ 3 + 3 a ^ 2 b x + 3 a b ^ 2 x
^ 2 + b ^ 3 x ^ 3))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(177)=354\).
time = 0.14, size = 459, normalized size = 2.45

method	result	size
norman	\(\frac {\frac {385 a^{7} d^{7}-1540 a^{6} b c \,d^{6}+2310 a^{5} b^{2} c^{2} d^{5}-1540 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}}{6 b^{8}}+\frac {d^{7} x^{7}}{4 b}+\frac {3 \left (35 a^{5} d^{7}-140 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}-140 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}-7 b^{5} c^{5} d^{2}\right ) x^{2}}{b^{6}}+\frac {\left (315 a^{6} d^{7}-1260 a^{5} b c \,d^{6}+1890 a^{4} b^{2} c^{2} d^{5}-1260 a^{3} b^{3} c^{3} d^{4}+315 a^{2} b^{4} c^{4} d^{3}-42 a \,b^{5} c^{5} d^{2}-7 b^{6} c^{6} d \right ) x}{2 b^{7}}-\frac {35 d^{4} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}+\frac {7 d^{5} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) x^{5}}{4 b^{3}}-\frac {7 d^{6} \left (a d -4 b c \right ) x^{6}}{12 b^{2}}}{\left (b x +a \right )^{3}}+\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{8}}\)	\(448\)
default	\(-\frac {d^{4} \left (-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {4}{3} a \,b^{2} d^{3} x^{3}-\frac {7}{3} b^{3} c \,d^{2} x^{3}-5 a^{2} b \,d^{3} x^{2}+14 a \,b^{2} c \,d^{2} x^{2}-\frac {21}{2} b^{3} c^{2} d \,x^{2}+20 a^{3} d^{3} x -70 a^{2} b c \,d^{2} x +84 a \,b^{2} c^{2} d x -35 b^{3} c^{3} x \right )}{b^{7}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{8} \left (b x +a \right )}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{3 b^{8} \left (b x +a \right )^{3}}\)	\(459\)
risch	\(\frac {d^{7} x^{4}}{4 b^{4}}-\frac {4 d^{7} a \,x^{3}}{3 b^{5}}+\frac {7 d^{6} c \,x^{3}}{3 b^{4}}+\frac {5 d^{7} a^{2} x^{2}}{b^{6}}-\frac {14 d^{6} a c \,x^{2}}{b^{5}}+\frac {21 d^{5} c^{2} x^{2}}{2 b^{4}}-\frac {20 d^{7} a^{3} x}{b^{7}}+\frac {70 d^{6} a^{2} c x}{b^{6}}-\frac {84 d^{5} a \,c^{2} x}{b^{5}}+\frac {35 d^{4} c^{3} x}{b^{4}}+\frac {\left (21 a^{5} b \,d^{7}-105 a^{4} b^{2} c \,d^{6}+210 a^{3} b^{3} c^{2} d^{5}-210 a^{2} b^{4} c^{3} d^{4}+105 a \,b^{5} c^{4} d^{3}-21 b^{6} c^{5} d^{2}\right ) x^{2}+\frac {7 d \left (11 a^{6} d^{6}-54 a^{5} b c \,d^{5}+105 a^{4} b^{2} c^{2} d^{4}-100 a^{3} b^{3} c^{3} d^{3}+45 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d -b^{6} c^{6}\right ) x}{2}+\frac {107 a^{7} d^{7}-518 a^{6} b c \,d^{6}+987 a^{5} b^{2} c^{2} d^{5}-910 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}}{6 b}}{b^{7} \left (b x +a \right )^{3}}+\frac {35 d^{7} \ln \left (b x +a \right ) a^{4}}{b^{8}}-\frac {140 d^{6} \ln \left (b x +a \right ) a^{3} c}{b^{7}}+\frac {210 d^{5} \ln \left (b x +a \right ) a^{2} c^{2}}{b^{6}}-\frac {140 d^{4} \ln \left (b x +a \right ) a \,c^{3}}{b^{5}}+\frac {35 d^{3} \ln \left (b x +a \right ) c^{4}}{b^{4}}\)	\(488\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^4/b^7*(-1/4*d^3*x^4*b^3+4/3*a*b^2*d^3*x^3-7/3*b^3*c*d^2*x^3-5*a^2*b*d^3*x^2+14*a*b^2*c*d^2*x^2-21/2*b^3*c^2
*d*x^2+20*a^3*d^3*x-70*a^2*b*c*d^2*x+84*a*b^2*c^2*d*x-35*b^3*c^3*x)+21/b^8*d^2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b
^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/(b*x+a)-7/2/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)/(b*x+a)^2+35/b^8*d^3*(a^4*d^4-4*a^3*b*c*d^3+6*
a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(b*x+a)-1/3/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)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (177) = 354\).
time = 0.28, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (177) = 354\).
time = 0.31, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (172) = 344\).
time = 22.14, 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} \cdot \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} \cdot \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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (177) = 354\).
time = 0.00, size = 502, normalized size = 2.68 \begin {gather*} \frac {\frac {1}{4} x^{4} d^{7} b^{12}-\frac {4}{3} x^{3} d^{7} b^{11} a+\frac {7}{3} x^{3} d^{6} c b^{12}+5 x^{2} d^{7} b^{10} a^{2}-14 x^{2} d^{6} c b^{11} a+\frac {21}{2} x^{2} d^{5} c^{2} b^{12}-20 x d^{7} b^{9} a^{3}+70 x d^{6} c b^{10} a^{2}-84 x d^{5} c^{2} b^{11} a+35 x d^{4} c^{3} b^{12}}{b^{16}}+\frac {\frac {1}{6} \left (\left (126 d^{7} b^{2} a^{5}-630 d^{6} b^{3} a^{4} c+1260 d^{5} b^{4} a^{3} c^{2}-1260 d^{4} b^{5} a^{2} c^{3}+630 d^{3} b^{6} a c^{4}-126 d^{2} b^{7} c^{5}\right ) x^{2}+\left (231 d^{7} b a^{6}-1134 d^{6} b^{2} a^{5} c+2205 d^{5} b^{3} a^{4} c^{2}-2100 d^{4} b^{4} a^{3} c^{3}+945 d^{3} b^{5} a^{2} c^{4}-126 d^{2} b^{6} a c^{5}-21 d b^{7} c^{6}\right ) x+107 d^{7} a^{7}-518 d^{6} b a^{6} c+987 d^{5} b^{2} a^{5} c^{2}-910 d^{4} b^{3} a^{4} c^{3}+385 d^{3} b^{4} a^{3} c^{4}-42 d^{2} b^{5} a^{2} c^{5}-7 d b^{6} a c^{6}-2 b^{7} c^{7}\right )}{b^{8} \left (x b+a\right )^{3}}+\frac {\left (35 d^{7} a^{4}-140 d^{6} c b a^{3}+210 d^{5} c^{2} b^{2} a^{2}-140 d^{4} c^{3} b^{3} a+35 d^{3} c^{4} b^{4}\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)^4,x)

[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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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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