Optimal. Leaf size=44 \[ \frac {(a+b x)^7 (A b-8 a B)}{56 a^2 x^7}-\frac {A (a+b x)^7}{8 a x^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 78, 37} \[ \frac {(a+b x)^7 (A b-8 a B)}{56 a^2 x^7}-\frac {A (a+b x)^7}{8 a x^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^9} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^9} \, dx\\ &=-\frac {A (a+b x)^7}{8 a x^8}+\frac {(-A b+8 a B) \int \frac {(a+b x)^6}{x^8} \, dx}{8 a}\\ &=-\frac {A (a+b x)^7}{8 a x^8}+\frac {(A b-8 a B) (a+b x)^7}{56 a^2 x^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.03, size = 123, normalized size = 2.80 \[ -\frac {a^6 (7 A+8 B x)+8 a^5 b x (6 A+7 B x)+28 a^4 b^2 x^2 (5 A+6 B x)+56 a^3 b^3 x^3 (4 A+5 B x)+70 a^2 b^4 x^4 (3 A+4 B x)+56 a b^5 x^5 (2 A+3 B x)+28 b^6 x^6 (A+2 B x)}{56 x^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 147, normalized size = 3.34 \[ -\frac {56 \, B b^{6} x^{7} + 7 \, A a^{6} + 28 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 56 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 70 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 56 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 28 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 8 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{56 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.15, size = 147, normalized size = 3.34 \[ -\frac {56 \, B b^{6} x^{7} + 168 \, B a b^{5} x^{6} + 28 \, A b^{6} x^{6} + 280 \, B a^{2} b^{4} x^{5} + 112 \, A a b^{5} x^{5} + 280 \, B a^{3} b^{3} x^{4} + 210 \, A a^{2} b^{4} x^{4} + 168 \, B a^{4} b^{2} x^{3} + 224 \, A a^{3} b^{3} x^{3} + 56 \, B a^{5} b x^{2} + 140 \, A a^{4} b^{2} x^{2} + 8 \, B a^{6} x + 48 \, A a^{5} b x + 7 \, A a^{6}}{56 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 128, normalized size = 2.91 \[ -\frac {B \,b^{6}}{x}-\frac {\left (A b +6 B a \right ) b^{5}}{2 x^{2}}-\frac {\left (2 A b +5 B a \right ) a \,b^{4}}{x^{3}}-\frac {5 \left (3 A b +4 B a \right ) a^{2} b^{3}}{4 x^{4}}-\frac {\left (4 A b +3 B a \right ) a^{3} b^{2}}{x^{5}}-\frac {A \,a^{6}}{8 x^{8}}-\frac {\left (5 A b +2 B a \right ) a^{4} b}{2 x^{6}}-\frac {\left (6 A b +B a \right ) a^{5}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.54, size = 147, normalized size = 3.34 \[ -\frac {56 \, B b^{6} x^{7} + 7 \, A a^{6} + 28 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 56 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 70 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 56 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 28 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 8 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{56 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.09, size = 141, normalized size = 3.20 \[ -\frac {x\,\left (\frac {B\,a^6}{7}+\frac {6\,A\,b\,a^5}{7}\right )+\frac {A\,a^6}{8}+x^2\,\left (B\,a^5\,b+\frac {5\,A\,a^4\,b^2}{2}\right )+x^5\,\left (5\,B\,a^2\,b^4+2\,A\,a\,b^5\right )+x^6\,\left (\frac {A\,b^6}{2}+3\,B\,a\,b^5\right )+x^3\,\left (3\,B\,a^4\,b^2+4\,A\,a^3\,b^3\right )+x^4\,\left (5\,B\,a^3\,b^3+\frac {15\,A\,a^2\,b^4}{4}\right )+B\,b^6\,x^7}{x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 5.52, size = 158, normalized size = 3.59 \[ \frac {- 7 A a^{6} - 56 B b^{6} x^{7} + x^{6} \left (- 28 A b^{6} - 168 B a b^{5}\right ) + x^{5} \left (- 112 A a b^{5} - 280 B a^{2} b^{4}\right ) + x^{4} \left (- 210 A a^{2} b^{4} - 280 B a^{3} b^{3}\right ) + x^{3} \left (- 224 A a^{3} b^{3} - 168 B a^{4} b^{2}\right ) + x^{2} \left (- 140 A a^{4} b^{2} - 56 B a^{5} b\right ) + x \left (- 48 A a^{5} b - 8 B a^{6}\right )}{56 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________