Optimal. Leaf size=347 \[ -\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2} \]
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Rubi [A] time = 0.60, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {794, 656, 648} \begin {gather*} -\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+3 c d g+13 c e f)}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int (d+e x)^{3/2} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx &=-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {3}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{13 c e^3}\\ &=-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac {(6 (2 c d-b e) (13 c e f+3 c d g-8 b e g)) \int \sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{143 c^2 e}\\ &=-\frac {4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac {\left (8 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{429 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}+\frac {\left (16 (2 c d-b e)^3 (13 c e f+3 c d g-8 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003 c^4 e}\\ &=-\frac {32 (2 c d-b e)^3 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15015 c^5 e^2 (d+e x)^{5/2}}-\frac {16 (2 c d-b e)^2 (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 c^4 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) (13 c e f+3 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 c^3 e^2 \sqrt {d+e x}}-\frac {2 (13 c e f+3 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 c^2 e^2}-\frac {2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 c e^2}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 264, normalized size = 0.76 \begin {gather*} -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (128 b^4 e^4 g-16 b^3 c e^3 (71 d g+13 e f+20 e g x)+8 b^2 c^2 e^2 \left (473 d^2 g+d e (221 f+315 g x)+5 e^2 x (13 f+14 g x)\right )-2 b c^3 e \left (2765 d^3 g+d^2 e (2743 f+3470 g x)+25 d e^2 x (78 f+77 g x)+35 e^3 x^2 (13 f+12 g x)\right )+c^4 \left (2754 d^4 g+d^3 e (6929 f+6885 g x)+5 d^2 e^2 x (1963 f+1659 g x)+35 d e^3 x^2 (169 f+141 g x)+105 e^4 x^3 (13 f+11 g x)\right )\right )}{15015 c^5 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.97, size = 401, normalized size = 1.16 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{5/2} \left (128 b^4 e^4 g-320 b^3 c e^3 g (d+e x)-816 b^3 c d e^3 g-208 b^3 c e^4 f+1824 b^2 c^2 d^2 e^2 g+520 b^2 c^2 e^3 f (d+e x)+1248 b^2 c^2 d e^3 f+560 b^2 c^2 e^2 g (d+e x)^2+1400 b^2 c^2 d e^2 g (d+e x)-1600 b c^3 d^3 e g-2496 b c^3 d^2 e^2 f-1760 b c^3 d^2 e g (d+e x)-910 b c^3 e^2 f (d+e x)^2-2080 b c^3 d e^2 f (d+e x)-840 b c^3 e g (d+e x)^3-1330 b c^3 d e g (d+e x)^2+384 c^4 d^4 g+1664 c^4 d^3 e f+480 c^4 d^3 g (d+e x)+2080 c^4 d^2 e f (d+e x)+420 c^4 d^2 g (d+e x)^2+1365 c^4 e f (d+e x)^3+1820 c^4 d e f (d+e x)^2+1155 c^4 g (d+e x)^4+315 c^4 d g (d+e x)^3\right )}{15015 c^5 e^2 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 678, normalized size = 1.95 \begin {gather*} -\frac {2 \, {\left (1155 \, c^{6} e^{6} g x^{6} + 105 \, {\left (13 \, c^{6} e^{6} f + {\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} g\right )} x^{5} + 35 \, {\left (13 \, {\left (7 \, c^{6} d e^{5} + 4 \, b c^{5} e^{6}\right )} f - {\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} g\right )} x^{4} - 5 \, {\left (13 \, {\left (10 \, c^{6} d^{2} e^{4} - 108 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} f + {\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} - 3 \, {\left (13 \, {\left (174 \, c^{6} d^{3} e^{3} - 236 \, b c^{5} d^{2} e^{4} - 17 \, b^{2} c^{4} d e^{5} + 2 \, b^{3} c^{3} e^{6}\right )} f + {\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} + 13 \, {\left (533 \, c^{6} d^{5} e - 1488 \, b c^{5} d^{4} e^{2} + 1513 \, b^{2} c^{4} d^{3} e^{3} - 710 \, b^{3} c^{3} d^{2} e^{4} + 168 \, b^{4} c^{2} d e^{5} - 16 \, b^{5} c e^{6}\right )} f + 2 \, {\left (1377 \, c^{6} d^{6} - 5519 \, b c^{5} d^{5} e + 8799 \, b^{2} c^{4} d^{4} e^{2} - 7117 \, b^{3} c^{3} d^{3} e^{3} + 3092 \, b^{4} c^{2} d^{2} e^{4} - 696 \, b^{5} c d e^{5} + 64 \, b^{6} e^{6}\right )} g - {\left (13 \, {\left (311 \, c^{6} d^{4} e^{2} - 100 \, b c^{5} d^{3} e^{3} - 279 \, b^{2} c^{4} d^{2} e^{4} + 76 \, b^{3} c^{3} d e^{5} - 8 \, b^{4} c^{2} e^{6}\right )} f - {\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15015 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 367, normalized size = 1.06 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (1155 g \,e^{4} x^{4} c^{4}-840 b \,c^{3} e^{4} g \,x^{3}+4935 c^{4} d \,e^{3} g \,x^{3}+1365 c^{4} e^{4} f \,x^{3}+560 b^{2} c^{2} e^{4} g \,x^{2}-3850 b \,c^{3} d \,e^{3} g \,x^{2}-910 b \,c^{3} e^{4} f \,x^{2}+8295 c^{4} d^{2} e^{2} g \,x^{2}+5915 c^{4} d \,e^{3} f \,x^{2}-320 b^{3} c \,e^{4} g x +2520 b^{2} c^{2} d \,e^{3} g x +520 b^{2} c^{2} e^{4} f x -6940 b \,c^{3} d^{2} e^{2} g x -3900 b \,c^{3} d \,e^{3} f x +6885 c^{4} d^{3} e g x +9815 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1136 b^{3} c d \,e^{3} g -208 b^{3} c \,e^{4} f +3784 b^{2} c^{2} d^{2} e^{2} g +1768 b^{2} c^{2} d \,e^{3} f -5530 b \,c^{3} d^{3} e g -5486 b \,c^{3} d^{2} e^{2} f +2754 c^{4} d^{4} g +6929 f \,d^{3} c^{4} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{\frac {3}{2}} c^{5} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.99, size = 676, normalized size = 1.95 \begin {gather*} -\frac {2 \, {\left (105 \, c^{5} e^{5} x^{5} + 533 \, c^{5} d^{5} - 1488 \, b c^{4} d^{4} e + 1513 \, b^{2} c^{3} d^{3} e^{2} - 710 \, b^{3} c^{2} d^{2} e^{3} + 168 \, b^{4} c d e^{4} - 16 \, b^{5} e^{5} + 35 \, {\left (7 \, c^{5} d e^{4} + 4 \, b c^{4} e^{5}\right )} x^{4} - 5 \, {\left (10 \, c^{5} d^{2} e^{3} - 108 \, b c^{4} d e^{4} - b^{2} c^{3} e^{5}\right )} x^{3} - 3 \, {\left (174 \, c^{5} d^{3} e^{2} - 236 \, b c^{4} d^{2} e^{3} - 17 \, b^{2} c^{3} d e^{4} + 2 \, b^{3} c^{2} e^{5}\right )} x^{2} - {\left (311 \, c^{5} d^{4} e - 100 \, b c^{4} d^{3} e^{2} - 279 \, b^{2} c^{3} d^{2} e^{3} + 76 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{1155 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} - \frac {2 \, {\left (1155 \, c^{6} e^{6} x^{6} + 2754 \, c^{6} d^{6} - 11038 \, b c^{5} d^{5} e + 17598 \, b^{2} c^{4} d^{4} e^{2} - 14234 \, b^{3} c^{3} d^{3} e^{3} + 6184 \, b^{4} c^{2} d^{2} e^{4} - 1392 \, b^{5} c d e^{5} + 128 \, b^{6} e^{6} + 105 \, {\left (25 \, c^{6} d e^{5} + 14 \, b c^{5} e^{6}\right )} x^{5} - 35 \, {\left (12 \, c^{6} d^{2} e^{4} - 154 \, b c^{5} d e^{5} - b^{2} c^{4} e^{6}\right )} x^{4} - 5 \, {\left (954 \, c^{6} d^{3} e^{3} - 1328 \, b c^{5} d^{2} e^{4} - 63 \, b^{2} c^{4} d e^{5} + 8 \, b^{3} c^{3} e^{6}\right )} x^{3} - 3 \, {\left (907 \, c^{6} d^{4} e^{2} - 560 \, b c^{5} d^{3} e^{3} - 473 \, b^{2} c^{4} d^{2} e^{4} + 142 \, b^{3} c^{3} d e^{5} - 16 \, b^{4} c^{2} e^{6}\right )} x^{2} + {\left (1377 \, c^{6} d^{5} e - 4142 \, b c^{5} d^{4} e^{2} + 4657 \, b^{2} c^{4} d^{3} e^{3} - 2460 \, b^{3} c^{3} d^{2} e^{4} + 632 \, b^{4} c^{2} d e^{5} - 64 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{15015 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 637, normalized size = 1.84 \begin {gather*} -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^2\,x^5\,\sqrt {d+e\,x}\,\left (14\,b\,e\,g+25\,c\,d\,g+13\,c\,e\,f\right )}{143}+\frac {2\,c\,e^3\,g\,x^6\,\sqrt {d+e\,x}}{13}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,g\,b^4\,c^2\,e^6-852\,g\,b^3\,c^3\,d\,e^5-156\,f\,b^3\,c^3\,e^6+2838\,g\,b^2\,c^4\,d^2\,e^4+1326\,f\,b^2\,c^4\,d\,e^5+3360\,g\,b\,c^5\,d^3\,e^3+18408\,f\,b\,c^5\,d^2\,e^4-5442\,g\,c^6\,d^4\,e^2-13572\,f\,c^6\,d^3\,e^3\right )}{15015\,c^5\,e^3}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1136\,g\,b^3\,c\,d\,e^3-208\,f\,b^3\,c\,e^4+3784\,g\,b^2\,c^2\,d^2\,e^2+1768\,f\,b^2\,c^2\,d\,e^3-5530\,g\,b\,c^3\,d^3\,e-5486\,f\,b\,c^3\,d^2\,e^2+2754\,g\,c^4\,d^4+6929\,f\,c^4\,d^3\,e\right )}{15015\,c^5\,e^3}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (g\,b^2\,e^2+154\,g\,b\,c\,d\,e+52\,f\,b\,c\,e^2-12\,g\,c^2\,d^2+91\,f\,c^2\,d\,e\right )}{429\,c}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-80\,g\,b^3\,c^3\,e^6+630\,g\,b^2\,c^4\,d\,e^5+130\,f\,b^2\,c^4\,e^6+13280\,g\,b\,c^5\,d^2\,e^4+14040\,f\,b\,c^5\,d\,e^5-9540\,g\,c^6\,d^3\,e^3-1300\,f\,c^6\,d^2\,e^4\right )}{15015\,c^5\,e^3}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (-64\,g\,b^4\,e^4+568\,g\,b^3\,c\,d\,e^3+104\,f\,b^3\,c\,e^4-1892\,g\,b^2\,c^2\,d^2\,e^2-884\,f\,b^2\,c^2\,d\,e^3+2765\,g\,b\,c^3\,d^3\,e+2743\,f\,b\,c^3\,d^2\,e^2-1377\,g\,c^4\,d^4+4043\,f\,c^4\,d^3\,e\right )}{15015\,c^4\,e^2}\right )}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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