Optimal. Leaf size=371 \[ -\frac {32 \sqrt {d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.55, antiderivative size = 371, 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 = {788, 656, 648} \begin {gather*} \frac {2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {32 \sqrt {d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(5 c e f+11 c d g-8 b e g) \int \frac {(d+e x)^{9/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(2 (5 c e f+11 c d g-8 b e g)) \int \frac {(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{5 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(8 (2 c d-b e) (5 c e f+11 c d g-8 b e g)) \int \frac {(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{15 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {16 (2 c d-b e) (5 c e f+11 c d g-8 b e g) (d+e x)^{3/2}}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (16 (2 c d-b e)^2 (5 c e f+11 c d g-8 b e g)\right ) \int \frac {(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}{15 c^4 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {32 (2 c d-b e)^2 (5 c e f+11 c d g-8 b e g) \sqrt {d+e x}}{15 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (2 c d-b e) (5 c e f+11 c d g-8 b e g) (d+e x)^{3/2}}{15 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 263, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {d+e x} \left (128 b^4 e^4 g-16 b^3 c e^3 (47 d g+5 e f-12 e g x)+24 b^2 c^2 e^2 \left (67 d^2 g+3 d e (5 f-13 g x)+e^2 x (2 g x-5 f)\right )-2 b c^3 e \left (741 d^3 g+3 d^2 e (85 f-246 g x)+3 d e^2 x (31 g x-70 f)+e^3 x^2 (15 f+4 g x)\right )+c^4 \left (498 d^4 g+9 d^3 e (25 f-83 g x)+3 d^2 e^2 x (61 g x-115 f)+d e^3 x^2 (75 f+23 g x)+e^4 x^3 (5 f+3 g x)\right )\right )}{15 c^5 e^2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 7.09, size = 401, normalized size = 1.08 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (128 b^4 e^4 g+192 b^3 c e^3 g (d+e x)-944 b^3 c d e^3 g-80 b^3 c e^4 f+2592 b^2 c^2 d^2 e^2 g-120 b^2 c^2 e^3 f (d+e x)+480 b^2 c^2 d e^3 f+48 b^2 c^2 e^2 g (d+e x)^2-1032 b^2 c^2 d e^2 g (d+e x)-3136 b c^3 d^3 e g-960 b c^3 d^2 e^2 f+1824 b c^3 d^2 e g (d+e x)-30 b c^3 e^2 f (d+e x)^2+480 b c^3 d e^2 f (d+e x)-8 b c^3 e g (d+e x)^3-162 b c^3 d e g (d+e x)^2+1408 c^4 d^4 g+640 c^4 d^3 e f-1056 c^4 d^3 g (d+e x)-480 c^4 d^2 e f (d+e x)+132 c^4 d^2 g (d+e x)^2+5 c^4 e f (d+e x)^3+60 c^4 d e f (d+e x)^2+3 c^4 g (d+e x)^4+11 c^4 d g (d+e x)^3\right )}{15 c^5 e^2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 428, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (3 \, c^{4} e^{4} g x^{4} + {\left (5 \, c^{4} e^{4} f + {\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} + 3 \, {\left (5 \, {\left (5 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f + {\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (45 \, c^{4} d^{3} e - 102 \, b c^{3} d^{2} e^{2} + 72 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f + 2 \, {\left (249 \, c^{4} d^{4} - 741 \, b c^{3} d^{3} e + 804 \, b^{2} c^{2} d^{2} e^{2} - 376 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g - 3 \, {\left (5 \, {\left (23 \, c^{4} d^{2} e^{2} - 28 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f + {\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\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}}{15 \, {\left (c^{7} e^{5} x^{3} + c^{7} d^{3} e^{2} - 2 \, b c^{6} d^{2} e^{3} + b^{2} c^{5} d e^{4} - {\left (c^{7} d e^{4} - 2 \, b c^{6} e^{5}\right )} x^{2} - {\left (c^{7} d^{2} e^{3} - b^{2} c^{5} e^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 0.99 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (3 g \,e^{4} x^{4} c^{4}-8 b \,c^{3} e^{4} g \,x^{3}+23 c^{4} d \,e^{3} g \,x^{3}+5 c^{4} e^{4} f \,x^{3}+48 b^{2} c^{2} e^{4} g \,x^{2}-186 b \,c^{3} d \,e^{3} g \,x^{2}-30 b \,c^{3} e^{4} f \,x^{2}+183 c^{4} d^{2} e^{2} g \,x^{2}+75 c^{4} d \,e^{3} f \,x^{2}+192 b^{3} c \,e^{4} g x -936 b^{2} c^{2} d \,e^{3} g x -120 b^{2} c^{2} e^{4} f x +1476 b \,c^{3} d^{2} e^{2} g x +420 b \,c^{3} d \,e^{3} f x -747 c^{4} d^{3} e g x -345 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -752 b^{3} c d \,e^{3} g -80 b^{3} c \,e^{4} f +1608 b^{2} c^{2} d^{2} e^{2} g +360 b^{2} c^{2} d \,e^{3} f -1482 b \,c^{3} d^{3} e g -510 b \,c^{3} d^{2} e^{2} f +498 c^{4} d^{4} g +225 c^{4} d^{3} e f \right ) \left (e x +d \right )^{\frac {5}{2}}}{15 \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} c^{5} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 363, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (c^{3} e^{3} x^{3} + 45 \, c^{3} d^{3} - 102 \, b c^{2} d^{2} e + 72 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 3 \, {\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{3 \, {\left (c^{5} e^{2} x - c^{5} d e + b c^{4} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (3 \, c^{4} e^{4} x^{4} + 498 \, c^{4} d^{4} - 1482 \, b c^{3} d^{3} e + 1608 \, b^{2} c^{2} d^{2} e^{2} - 752 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + {\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} - 3 \, {\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{15 \, {\left (c^{6} e^{3} x - c^{6} d e^{2} + b c^{5} e^{3}\right )} \sqrt {-c e x + c d - b e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.29, size = 435, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (256\,g\,b^4\,e^4-1504\,g\,b^3\,c\,d\,e^3-160\,f\,b^3\,c\,e^4+3216\,g\,b^2\,c^2\,d^2\,e^2+720\,f\,b^2\,c^2\,d\,e^3-2964\,g\,b\,c^3\,d^3\,e-1020\,f\,b\,c^3\,d^2\,e^2+996\,g\,c^4\,d^4+450\,f\,c^4\,d^3\,e\right )}{15\,c^7\,e^5}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-62\,g\,b\,c\,d\,e-10\,f\,b\,c\,e^2+61\,g\,c^2\,d^2+25\,f\,c^2\,d\,e\right )}{5\,c^5\,e^3}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (23\,c\,d\,g-8\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^4\,e^2}+\frac {2\,g\,x^4\,\sqrt {d+e\,x}}{5\,c^3\,e}-\frac {x\,\sqrt {d+e\,x}\,\left (-384\,g\,b^3\,c\,e^4+1872\,g\,b^2\,c^2\,d\,e^3+240\,f\,b^2\,c^2\,e^4-2952\,g\,b\,c^3\,d^2\,e^2-840\,f\,b\,c^3\,d\,e^3+1494\,g\,c^4\,d^3\,e+690\,f\,c^4\,d^2\,e^2\right )}{15\,c^7\,e^5}\right )}{x^3+\frac {x\,\left (15\,b^2\,c^5\,e^5-15\,c^7\,d^2\,e^3\right )}{15\,c^7\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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