Optimal. Leaf size=347 \[ -\frac {32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{231 c^3 e^2}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2} \]
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Rubi [A] time = 0.62, 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 )^{3/2} (-8 b e g+5 c d g+11 c e f)}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{231 c^3 e^2}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+5 c d g+11 c e f)}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 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)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx &=-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}-\frac {\left (2 \left (\frac {3}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int (d+e x)^{5/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{11 c e^3}\\ &=-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}+\frac {(2 (2 c d-b e) (11 c e f+5 c d g-8 b e g)) \int (d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{33 c^2 e}\\ &=-\frac {4 (2 c d-b e) (11 c e f+5 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 )^{3/2}}{231 c^3 e^2}-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}+\frac {\left (8 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g)\right ) \int \sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{231 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) (11 c e f+5 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 )^{3/2}}{231 c^3 e^2}-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}+\frac {\left (16 (2 c d-b e)^3 (11 c e f+5 c d g-8 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx}{1155 c^4 e}\\ &=-\frac {32 (2 c d-b e)^3 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) (11 c e f+5 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 )^{3/2}}{231 c^3 e^2}-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 262, normalized size = 0.76 \begin {gather*} \frac {2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)} \left (128 b^4 e^4 g-16 b^3 c e^3 (65 d g+11 e f+12 e g x)+24 b^2 c^2 e^2 \left (131 d^2 g+d e (55 f+57 g x)+e^2 x (11 f+10 g x)\right )-2 b c^3 e \left (2071 d^3 g+3 d^2 e (583 f+558 g x)+3 d e^2 x (286 f+245 g x)+5 e^3 x^2 (33 f+28 g x)\right )+c^4 \left (1910 d^4 g+d^3 e (3509 f+2865 g x)+3 d^2 e^2 x (1177 f+905 g x)+5 d e^3 x^2 (363 f+287 g x)+35 e^4 x^3 (11 f+9 g x)\right )\right )}{3465 c^5 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.88, 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 )^{3/2} \left (128 b^4 e^4 g-192 b^3 c e^3 g (d+e x)-848 b^3 c d e^3 g-176 b^3 c e^4 f+2016 b^2 c^2 d^2 e^2 g+264 b^2 c^2 e^3 f (d+e x)+1056 b^2 c^2 d e^3 f+240 b^2 c^2 e^2 g (d+e x)^2+888 b^2 c^2 d e^2 g (d+e x)-1984 b c^3 d^3 e g-2112 b c^3 d^2 e^2 f-1248 b c^3 d^2 e g (d+e x)-330 b c^3 e^2 f (d+e x)^2-1056 b c^3 d e^2 f (d+e x)-280 b c^3 e g (d+e x)^3-630 b c^3 d e g (d+e x)^2+640 c^4 d^4 g+1408 c^4 d^3 e f+480 c^4 d^3 g (d+e x)+1056 c^4 d^2 e f (d+e x)+300 c^4 d^2 g (d+e x)^2+385 c^4 e f (d+e x)^3+660 c^4 d e f (d+e x)^2+315 c^4 g (d+e x)^4+175 c^4 d g (d+e x)^3\right )}{3465 c^5 e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 499, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (315 \, c^{5} e^{5} g x^{5} + 35 \, {\left (11 \, c^{5} e^{5} f + {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \, {\left (11 \, {\left (26 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} f + {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 3 \, {\left (11 \, {\left (52 \, c^{5} d^{2} e^{3} + 13 \, b c^{4} d e^{4} - 2 \, b^{2} c^{3} e^{5}\right )} f + {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \, {\left (319 \, c^{5} d^{4} e - 637 \, b c^{4} d^{3} e^{2} + 438 \, b^{2} c^{3} d^{2} e^{3} - 136 \, b^{3} c^{2} d e^{4} + 16 \, b^{4} c e^{5}\right )} f - 2 \, {\left (955 \, c^{5} d^{5} - 3026 \, b c^{4} d^{4} e + 3643 \, b^{2} c^{3} d^{3} e^{2} - 2092 \, b^{3} c^{2} d^{2} e^{3} + 584 \, b^{4} c d e^{4} - 64 \, b^{5} e^{5}\right )} g - {\left (11 \, {\left (2 \, c^{5} d^{3} e^{2} - 159 \, b c^{4} d^{2} e^{3} + 60 \, b^{2} c^{3} d e^{4} - 8 \, b^{3} c^{2} e^{5}\right )} f + {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\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}}{3465 \, {\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 \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (e x + d\right )}^{\frac {5}{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 (315 g \,e^{4} x^{4} c^{4}-280 b \,c^{3} e^{4} g \,x^{3}+1435 c^{4} d \,e^{3} g \,x^{3}+385 c^{4} e^{4} f \,x^{3}+240 b^{2} c^{2} e^{4} g \,x^{2}-1470 b \,c^{3} d \,e^{3} g \,x^{2}-330 b \,c^{3} e^{4} f \,x^{2}+2715 c^{4} d^{2} e^{2} g \,x^{2}+1815 c^{4} d \,e^{3} f \,x^{2}-192 b^{3} c \,e^{4} g x +1368 b^{2} c^{2} d \,e^{3} g x +264 b^{2} c^{2} e^{4} f x -3348 b \,c^{3} d^{2} e^{2} g x -1716 b \,c^{3} d \,e^{3} f x +2865 c^{4} d^{3} e g x +3531 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1040 b^{3} c d \,e^{3} g -176 b^{3} c \,e^{4} f +3144 b^{2} c^{2} d^{2} e^{2} g +1320 b^{2} c^{2} d \,e^{3} f -4142 b \,c^{3} d^{3} e g -3498 b \,c^{3} d^{2} e^{2} f +1910 c^{4} d^{4} g +3509 f \,d^{3} c^{4} e \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{3465 \sqrt {e x +d}\, c^{5} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 501, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} - 319 \, c^{4} d^{4} + 637 \, b c^{3} d^{3} e - 438 \, b^{2} c^{2} d^{2} e^{2} + 136 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \, {\left (26 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (52 \, c^{4} d^{2} e^{2} + 13 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} - {\left (2 \, c^{4} d^{3} e - 159 \, b c^{3} d^{2} e^{2} + 60 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{315 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} + \frac {2 \, {\left (315 \, c^{5} e^{5} x^{5} - 1910 \, c^{5} d^{5} + 6052 \, b c^{4} d^{4} e - 7286 \, b^{2} c^{3} d^{3} e^{2} + 4184 \, b^{3} c^{2} d^{2} e^{3} - 1168 \, b^{4} c d e^{4} + 128 \, b^{5} e^{5} + 35 \, {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} x^{4} + 5 \, {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} x^{2} - {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{3465 \, {\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 = 3.09, size = 501, normalized size = 1.44 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (-8\,g\,b^2\,e^2+49\,g\,b\,c\,d\,e+11\,f\,b\,c\,e^2+256\,g\,c^2\,d^2+286\,f\,c^2\,d\,e\right )}{693\,c^2}+\frac {2\,e^2\,g\,x^5\,\sqrt {d+e\,x}}{11}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1040\,g\,b^3\,c\,d\,e^3-176\,f\,b^3\,c\,e^4+3144\,g\,b^2\,c^2\,d^2\,e^2+1320\,f\,b^2\,c^2\,d\,e^3-4142\,g\,b\,c^3\,d^3\,e-3498\,f\,b\,c^3\,d^2\,e^2+1910\,g\,c^4\,d^4+3509\,f\,c^4\,d^3\,e\right )}{3465\,c^5\,e^3}-\frac {x\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,c\,e^5-1040\,g\,b^3\,c^2\,d\,e^4-176\,f\,b^3\,c^2\,e^5+3144\,g\,b^2\,c^3\,d^2\,e^3+1320\,f\,b^2\,c^3\,d\,e^4-4142\,g\,b\,c^4\,d^3\,e^2-3498\,f\,b\,c^4\,d^2\,e^3+1910\,g\,c^5\,d^4\,e+44\,f\,c^5\,d^3\,e^2\right )}{3465\,c^5\,e^3}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,g\,b^3\,c^2\,e^5-684\,g\,b^2\,c^3\,d\,e^4-132\,f\,b^2\,c^3\,e^5+1674\,g\,b\,c^4\,d^2\,e^3+858\,f\,b\,c^4\,d\,e^4+300\,g\,c^5\,d^3\,e^2+3432\,f\,c^5\,d^2\,e^3\right )}{3465\,c^5\,e^3}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (b\,e\,g+32\,c\,d\,g+11\,c\,e\,f\right )}{99\,c}\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 \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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