Optimal. Leaf size=164 \[ -\frac {512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac {256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac {64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac {4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {656, 648} \[ -\frac {512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac {256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac {64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac {4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 648
Rule 656
Rubi steps
\begin {align*} \int x^{7/2} \left (b x+c x^2\right )^{3/2} \, dx &=\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {(2 b) \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx}{3 c}\\ &=-\frac {4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}+\frac {\left (16 b^2\right ) \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx}{39 c^2}\\ &=\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac {4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {\left (32 b^3\right ) \int \sqrt {x} \left (b x+c x^2\right )^{3/2} \, dx}{143 c^3}\\ &=-\frac {64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac {4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}+\frac {\left (128 b^4\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{\sqrt {x}} \, dx}{1287 c^4}\\ &=\frac {256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac {64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac {4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}-\frac {\left (256 b^5\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{9009 c^5}\\ &=-\frac {512 b^5 \left (b x+c x^2\right )^{5/2}}{45045 c^6 x^{5/2}}+\frac {256 b^4 \left (b x+c x^2\right )^{5/2}}{9009 c^5 x^{3/2}}-\frac {64 b^3 \left (b x+c x^2\right )^{5/2}}{1287 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{5/2}}{429 c^3}-\frac {4 b x^{3/2} \left (b x+c x^2\right )^{5/2}}{39 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{5/2}}{15 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 75, normalized size = 0.46 \[ \frac {2 (x (b+c x))^{5/2} \left (-256 b^5+640 b^4 c x-1120 b^3 c^2 x^2+1680 b^2 c^3 x^3-2310 b c^4 x^4+3003 c^5 x^5\right )}{45045 c^6 x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 93, normalized size = 0.57 \[ \frac {2 \, {\left (3003 \, c^{7} x^{7} + 3696 \, b c^{6} x^{6} + 63 \, b^{2} c^{5} x^{5} - 70 \, b^{3} c^{4} x^{4} + 80 \, b^{4} c^{3} x^{3} - 96 \, b^{5} c^{2} x^{2} + 128 \, b^{6} c x - 256 \, b^{7}\right )} \sqrt {c x^{2} + b x}}{45045 \, c^{6} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 182, normalized size = 1.11 \[ -\frac {2}{45045} \, c {\left (\frac {1024 \, b^{\frac {15}{2}}}{c^{7}} - \frac {3003 \, {\left (c x + b\right )}^{\frac {15}{2}} - 20790 \, {\left (c x + b\right )}^{\frac {13}{2}} b + 61425 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{2} - 100100 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{3} + 96525 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{4} - 54054 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{6}}{c^{7}}\right )} + \frac {2}{9009} \, b {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 77, normalized size = 0.47 \[ -\frac {2 \left (c x +b \right ) \left (-3003 c^{5} x^{5}+2310 b \,x^{4} c^{4}-1680 b^{2} x^{3} c^{3}+1120 b^{3} x^{2} c^{2}-640 b^{4} x c +256 b^{5}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{45045 c^{6} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.48, size = 168, normalized size = 1.02 \[ \frac {2 \, {\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 5 \, {\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5}\right )} \sqrt {c x + b}}{45045 \, c^{6} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{7/2}\,{\left (c\,x^2+b\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________