Optimal. Leaf size=360 \[ -\frac {32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{45045 e^2 (d+e x)^7 (2 c d-b e)^5}-\frac {16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{6435 e^2 (d+e x)^8 (2 c d-b e)^4}-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{715 e^2 (d+e x)^9 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{195 e^2 (d+e x)^{10} (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)} \]
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Rubi [A] time = 0.58, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {792, 658, 650} \begin {gather*} -\frac {32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{45045 e^2 (d+e x)^7 (2 c d-b e)^5}-\frac {16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{6435 e^2 (d+e x)^8 (2 c d-b e)^4}-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{715 e^2 (d+e x)^9 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{195 e^2 (d+e x)^{10} (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rule 792
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}+\frac {(8 c e f+22 c d g-15 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx}{15 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac {2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}+\frac {(2 c (8 c e f+22 c d g-15 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx}{65 e (2 c d-b e)^2}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac {2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}-\frac {4 c (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 e^2 (2 c d-b e)^3 (d+e x)^9}+\frac {\left (8 c^2 (8 c e f+22 c d g-15 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx}{715 e (2 c d-b e)^3}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac {2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}-\frac {4 c (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 e^2 (2 c d-b e)^3 (d+e x)^9}-\frac {16 c^2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 e^2 (2 c d-b e)^4 (d+e x)^8}+\frac {\left (16 c^3 (8 c e f+22 c d g-15 b e g)\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx}{6435 e (2 c d-b e)^4}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac {2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}-\frac {4 c (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 e^2 (2 c d-b e)^3 (d+e x)^9}-\frac {16 c^2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 e^2 (2 c d-b e)^4 (d+e x)^8}-\frac {32 c^3 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{45045 e^2 (2 c d-b e)^5 (d+e x)^7}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 351, normalized size = 0.98 \begin {gather*} -\frac {2 (b e-c d+c e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} \left (231 b^4 e^4 (2 d g+13 e f+15 e g x)-42 b^3 c e^3 \left (89 d^2 g+d e (616 f+706 g x)+e^2 x (44 f+45 g x)\right )+84 b^2 c^2 e^2 \left (133 d^3 g+6 d^2 e (167 f+189 g x)+3 d e^2 x (52 f+51 g x)+2 e^3 x^2 (6 f+5 g x)\right )-8 b c^3 e \left (1801 d^4 g+2 d^3 e (7672 f+8481 g x)+3 d^2 e^2 x (1316 f+1201 g x)+4 d e^3 x^2 (168 f+121 g x)+2 e^4 x^3 (28 f+15 g x)\right )+16 c^4 \left (407 d^5 g+d^4 e (4243 f+4477 g x)+11 d^3 e^2 x (148 f+117 g x)+2 d^2 e^3 x^2 (234 f+121 g x)+22 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{45045 e^2 (d+e x)^8 (b e-2 c d)^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.61, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [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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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 = 0.06, size = 564, normalized size = 1.57 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-240 b \,c^{3} e^{5} g \,x^{4}+352 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}+840 b^{2} c^{2} e^{5} g \,x^{3}-3872 b \,c^{3} d \,e^{4} g \,x^{3}-448 b \,c^{3} e^{5} f \,x^{3}+3872 c^{4} d^{2} e^{3} g \,x^{3}+1408 c^{4} d \,e^{4} f \,x^{3}-1890 b^{3} c \,e^{5} g \,x^{2}+12852 b^{2} c^{2} d \,e^{4} g \,x^{2}+1008 b^{2} c^{2} e^{5} f \,x^{2}-28824 b \,c^{3} d^{2} e^{3} g \,x^{2}-5376 b \,c^{3} d \,e^{4} f \,x^{2}+20592 c^{4} d^{3} e^{2} g \,x^{2}+7488 c^{4} d^{2} e^{3} f \,x^{2}+3465 b^{4} e^{5} g x -29652 b^{3} c d \,e^{4} g x -1848 b^{3} c \,e^{5} f x +95256 b^{2} c^{2} d^{2} e^{3} g x +13104 b^{2} c^{2} d \,e^{4} f x -135696 b \,c^{3} d^{3} e^{2} g x -31584 b \,c^{3} d^{2} e^{3} f x +71632 c^{4} d^{4} e g x +26048 c^{4} d^{3} e^{2} f x +462 b^{4} d \,e^{4} g +3003 b^{4} e^{5} f -3738 b^{3} c \,d^{2} e^{3} g -25872 b^{3} c d \,e^{4} f +11172 b^{2} c^{2} d^{3} e^{2} g +84168 b^{2} c^{2} d^{2} e^{3} f -14408 b \,c^{3} d^{4} e g -122752 b \,c^{3} d^{3} e^{2} f +6512 c^{4} d^{5} g +67888 c^{4} d^{4} e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{10} \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} c^{2} d^{2} e^{3}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 c^{5} d^{5}\right ) e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \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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