Optimal. Leaf size=185 \[ \frac{7 d^6 (a+b x)^4 (b c-a d)}{4 b^8}+\frac{7 d^5 (a+b x)^3 (b c-a d)^2}{b^8}+\frac{35 d^4 (a+b x)^2 (b c-a d)^3}{2 b^8}+\frac{35 d^3 x (b c-a d)^4}{b^7}+\frac{21 d^2 (b c-a d)^5 \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{b^8 (a+b x)}-\frac{(b c-a d)^7}{2 b^8 (a+b x)^2}+\frac{d^7 (a+b x)^5}{5 b^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21712, antiderivative size = 185, normalized size of antiderivative = 1., 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} \[ \frac{7 d^6 (a+b x)^4 (b c-a d)}{4 b^8}+\frac{7 d^5 (a+b x)^3 (b c-a d)^2}{b^8}+\frac{35 d^4 (a+b x)^2 (b c-a d)^3}{2 b^8}+\frac{35 d^3 x (b c-a d)^4}{b^7}+\frac{21 d^2 (b c-a d)^5 \log (a+b x)}{b^8}-\frac{7 d (b c-a d)^6}{b^8 (a+b x)}-\frac{(b c-a d)^7}{2 b^8 (a+b x)^2}+\frac{d^7 (a+b x)^5}{5 b^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
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*}
Mathematica [B] time = 0.129932, size = 389, normalized size = 2.1 \[ \frac{7 a^2 b^5 d^2 \left (-550 c^3 d^2 x^2+200 c^2 d^3 x^3-200 c^4 d x+90 c^5+25 c d^4 x^4+2 d^5 x^5\right )-35 a^3 b^4 d^3 \left (-126 c^2 d^2 x^2-20 c^3 d x+50 c^4+20 c d^3 x^3+d^4 x^4\right )+70 a^4 b^3 d^4 \left (6 c^2 d x+35 c^3-34 c d^2 x^2+2 d^3 x^3\right )+10 a^5 b^2 d^5 \left (-189 c^2-56 c d x+50 d^2 x^2\right )+10 a^6 b d^6 (77 c+16 d x)-130 a^7 d^7-7 a b^6 d \left (-200 c^4 d^2 x^2+200 c^3 d^3 x^3+50 c^2 d^4 x^4-120 c^5 d x+10 c^6+10 c d^5 x^5+d^6 x^6\right )-420 d^2 (a+b x)^2 (a d-b c)^5 \log (a+b x)+b^7 \left (700 c^4 d^3 x^3+350 c^3 d^4 x^4+140 c^2 d^5 x^5-140 c^6 d x-10 c^7+35 c d^6 x^6+4 d^7 x^7\right )}{20 b^8 (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.011, size = 599, normalized size = 3.2 \begin{align*} -{\frac{{c}^{7}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{{d}^{7}{x}^{5}}{5\,{b}^{3}}}-7\,{\frac{{a}^{6}{d}^{7}}{{b}^{8} \left ( bx+a \right ) }}-{\frac{3\,{d}^{7}{x}^{4}a}{4\,{b}^{4}}}+{\frac{{a}^{7}{d}^{7}}{2\,{b}^{8} \left ( bx+a \right ) ^{2}}}-21\,{\frac{{d}^{7}\ln \left ( bx+a \right ){a}^{5}}{{b}^{8}}}+21\,{\frac{{d}^{2}\ln \left ( bx+a \right ){c}^{5}}{{b}^{3}}}-7\,{\frac{d{c}^{6}}{{b}^{2} \left ( bx+a \right ) }}+35\,{\frac{{c}^{4}{d}^{3}x}{{b}^{3}}}+{\frac{35\,{d}^{4}{x}^{2}{c}^{3}}{2\,{b}^{3}}}+15\,{\frac{{a}^{4}{d}^{7}x}{{b}^{7}}}-5\,{\frac{{d}^{7}{x}^{2}{a}^{3}}{{b}^{6}}}+{\frac{7\,{d}^{6}{x}^{4}c}{4\,{b}^{3}}}+2\,{\frac{{d}^{7}{x}^{3}{a}^{2}}{{b}^{5}}}+7\,{\frac{{d}^{5}{x}^{3}{c}^{2}}{{b}^{3}}}+21\,{\frac{{d}^{6}{x}^{2}{a}^{2}c}{{b}^{5}}}+210\,{\frac{{d}^{4}\ln \left ( bx+a \right ){a}^{2}{c}^{3}}{{b}^{5}}}-105\,{\frac{{d}^{3}\ln \left ( bx+a \right ) a{c}^{4}}{{b}^{4}}}+42\,{\frac{{a}^{5}c{d}^{6}}{{b}^{7} \left ( bx+a \right ) }}-105\,{\frac{{d}^{5}{a}^{4}{c}^{2}}{{b}^{6} \left ( bx+a \right ) }}+140\,{\frac{{a}^{3}{c}^{3}{d}^{4}}{{b}^{5} \left ( bx+a \right ) }}-105\,{\frac{{a}^{2}{c}^{4}{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}+42\,{\frac{a{c}^{5}{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}-{\frac{7\,{a}^{6}c{d}^{6}}{2\,{b}^{7} \left ( bx+a \right ) ^{2}}}+{\frac{21\,{a}^{5}{c}^{2}{d}^{5}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-{\frac{35\,{a}^{4}{c}^{3}{d}^{4}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{35\,{a}^{3}{c}^{4}{d}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{21\,{a}^{2}{c}^{5}{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{7\,a{c}^{6}d}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{63\,{d}^{5}{x}^{2}a{c}^{2}}{2\,{b}^{4}}}-70\,{\frac{{a}^{3}c{d}^{6}x}{{b}^{6}}}+126\,{\frac{{a}^{2}{c}^{2}{d}^{5}x}{{b}^{5}}}-105\,{\frac{a{c}^{3}{d}^{4}x}{{b}^{4}}}-7\,{\frac{{d}^{6}{x}^{3}ac}{{b}^{4}}}-210\,{\frac{{d}^{5}\ln \left ( bx+a \right ){a}^{3}{c}^{2}}{{b}^{6}}}+105\,{\frac{{d}^{6}\ln \left ( bx+a \right ){a}^{4}c}{{b}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.02106, size = 639, normalized size = 3.45 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.38854, size = 1445, normalized size = 7.81 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 3.63683, size = 437, normalized size = 2.36 \begin{align*} - \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{x^{4} \left (3 a d^{7} - 7 b c d^{6}\right )}{4 b^{4}} + \frac{x^{3} \left (2 a^{2} d^{7} - 7 a b c d^{6} + 7 b^{2} c^{2} d^{5}\right )}{b^{5}} - \frac{x^{2} \left (10 a^{3} d^{7} - 42 a^{2} b c d^{6} + 63 a b^{2} c^{2} d^{5} - 35 b^{3} c^{3} d^{4}\right )}{2 b^{6}} + \frac{x \left (15 a^{4} d^{7} - 70 a^{3} b c d^{6} + 126 a^{2} b^{2} c^{2} d^{5} - 105 a b^{3} c^{3} d^{4} + 35 b^{4} c^{4} d^{3}\right )}{b^{7}} - \frac{21 d^{2} \left (a d - b c\right )^{5} \log{\left (a + b x \right )}}{b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.06849, size = 644, normalized size = 3.48 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]