Optimal. Leaf size=125 \[ -\frac {16 c^2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{3465 b^4 x^5}+\frac {8 c \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{693 b^3 x^6}-\frac {2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{99 b^2 x^7}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8} \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 658, 650} \begin {gather*} -\frac {16 c^2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{3465 b^4 x^5}+\frac {8 c \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{693 b^3 x^6}-\frac {2 \left (b x+c x^2\right )^{5/2} (11 b B-6 A c)}{99 b^2 x^7}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 650
Rule 658
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^8} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}+\frac {\left (2 \left (-8 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^7} \, dx}{11 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}-\frac {(4 c (11 b B-6 A c)) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx}{99 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}+\frac {8 c (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{693 b^3 x^6}+\frac {\left (8 c^2 (11 b B-6 A c)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{693 b^3}\\ &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{99 b^2 x^7}+\frac {8 c (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{693 b^3 x^6}-\frac {16 c^2 (11 b B-6 A c) \left (b x+c x^2\right )^{5/2}}{3465 b^4 x^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 79, normalized size = 0.63 \begin {gather*} -\frac {2 (x (b+c x))^{5/2} \left (3 A \left (105 b^3-70 b^2 c x+40 b c^2 x^2-16 c^3 x^3\right )+11 b B x \left (35 b^2-20 b c x+8 c^2 x^2\right )\right )}{3465 b^4 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.46, size = 132, normalized size = 1.06 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-315 A b^5-420 A b^4 c x-15 A b^3 c^2 x^2+18 A b^2 c^3 x^3-24 A b c^4 x^4+48 A c^5 x^5-385 b^5 B x-550 b^4 B c x^2-33 b^3 B c^2 x^3+44 b^2 B c^3 x^4-88 b B c^4 x^5\right )}{3465 b^4 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 130, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (315 \, A b^{5} + 8 \, {\left (11 \, B b c^{4} - 6 \, A c^{5}\right )} x^{5} - 4 \, {\left (11 \, B b^{2} c^{3} - 6 \, A b c^{4}\right )} x^{4} + 3 \, {\left (11 \, B b^{3} c^{2} - 6 \, A b^{2} c^{3}\right )} x^{3} + 5 \, {\left (110 \, B b^{4} c + 3 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} + 12 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.40, size = 431, normalized size = 3.45 \begin {gather*} \frac {2 \, {\left (4620 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B c^{3} + 17325 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b c^{\frac {5}{2}} + 6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A c^{\frac {7}{2}} + 28413 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 30492 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b c^{3} + 25410 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac {3}{2}} + 58905 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac {5}{2}} + 12870 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{4} c + 63855 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{5} \sqrt {c} + 41580 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{6} + 16170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{5} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{6} \sqrt {c} + 315 \, A b^{7}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 86, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-48 A \,c^{3} x^{3}+88 B b \,c^{2} x^{3}+120 A b \,c^{2} x^{2}-220 B \,b^{2} c \,x^{2}-210 A \,b^{2} c x +385 B \,b^{3} x +315 A \,b^{3}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3465 b^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.96, size = 268, normalized size = 2.14 \begin {gather*} -\frac {16 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{3} x} + \frac {32 \, \sqrt {c x^{2} + b x} A c^{5}}{1155 \, b^{4} x} + \frac {8 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{2} x^{2}} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{4}}{1155 \, b^{3} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b x^{3}} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{3}}{385 \, b^{2} x^{3}} + \frac {\sqrt {c x^{2} + b x} B c}{63 \, x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{2}}{231 \, b x^{4}} + \frac {\sqrt {c x^{2} + b x} B b}{9 \, x^{5}} + \frac {\sqrt {c x^{2} + b x} A c}{132 \, x^{5}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{3 \, x^{6}} + \frac {3 \, \sqrt {c x^{2} + b x} A b}{44 \, x^{6}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{4 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.88, size = 234, normalized size = 1.87 \begin {gather*} \frac {4\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{385\,b^2\,x^3}-\frac {8\,A\,c\,\sqrt {c\,x^2+b\,x}}{33\,x^5}-\frac {2\,B\,b\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {20\,B\,c\,\sqrt {c\,x^2+b\,x}}{63\,x^4}-\frac {2\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{231\,b\,x^4}-\frac {2\,A\,b\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {16\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^2}+\frac {32\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{1155\,b^4\,x}-\frac {2\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b\,x^3}+\frac {8\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{315\,b^2\,x^2}-\frac {16\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{315\,b^3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________