Optimal. Leaf size=360 \[ -\frac {32 c^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x) (2 c d-b e)^5}-\frac {16 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x)^2 (2 c d-b e)^4}-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{63 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (d+e x)^5 (2 c d-b e)} \]
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Rubi [A] time = 0.56, 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 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x) (2 c d-b e)^5}-\frac {16 c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x)^2 (2 c d-b e)^4}-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{63 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (d+e x)^5 (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}{(d+e x)^5 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}+\frac {(8 c e f+10 c d g-9 b e g) \int \frac {1}{(d+e x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{9 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}+\frac {(2 c (8 c e f+10 c d g-9 b e g)) \int \frac {1}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{21 e (2 c d-b e)^2}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}+\frac {\left (8 c^2 (8 c e f+10 c d g-9 b e g)\right ) \int \frac {1}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{105 e (2 c d-b e)^3}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}-\frac {16 c^2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^4 (d+e x)^2}+\frac {\left (16 c^3 (8 c e f+10 c d g-9 b e g)\right ) \int \frac {1}{(d+e x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{315 e (2 c d-b e)^4}\\ &=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}-\frac {16 c^2 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^4 (d+e x)^2}-\frac {32 c^3 (8 c e f+10 c d g-9 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^5 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 348, normalized size = 0.97 \begin {gather*} -\frac {2 (b e-c d+c e x) \left (5 b^4 e^4 (2 d g+7 e f+9 e g x)-2 b^3 c e^3 \left (47 d^2 g+2 d e (80 f+107 g x)+e^2 x (20 f+27 g x)\right )+12 b^2 c^2 e^2 \left (29 d^3 g+2 d^2 e (47 f+67 g x)+d e^2 x (28 f+41 g x)+2 e^3 x^2 (2 f+3 g x)\right )-8 b c^3 e \left (83 d^4 g+d^3 e (232 f+390 g x)+3 d^2 e^2 x (44 f+83 g x)+4 d e^3 x^2 (12 f+25 g x)+2 e^4 x^3 (4 f+9 g x)\right )+16 c^4 \left (25 d^5 g+d^4 e (83 f+125 g x)+5 d^3 e^2 x (20 f+21 g x)+2 d^2 e^3 x^2 (42 f+25 g x)+10 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{315 e^2 (d+e x)^4 (b e-2 c d)^5 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 111.18, size = 13927, normalized size = 38.69 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 564, normalized size = 1.57 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-144 b \,c^{3} e^{5} g \,x^{4}+160 c^{4} d \,e^{4} g \,x^{4}+128 c^{4} e^{5} f \,x^{4}+72 b^{2} c^{2} e^{5} g \,x^{3}-800 b \,c^{3} d \,e^{4} g \,x^{3}-64 b \,c^{3} e^{5} f \,x^{3}+800 c^{4} d^{2} e^{3} g \,x^{3}+640 c^{4} d \,e^{4} f \,x^{3}-54 b^{3} c \,e^{5} g \,x^{2}+492 b^{2} c^{2} d \,e^{4} g \,x^{2}+48 b^{2} c^{2} e^{5} f \,x^{2}-1992 b \,c^{3} d^{2} e^{3} g \,x^{2}-384 b \,c^{3} d \,e^{4} f \,x^{2}+1680 c^{4} d^{3} e^{2} g \,x^{2}+1344 c^{4} d^{2} e^{3} f \,x^{2}+45 b^{4} e^{5} g x -428 b^{3} c d \,e^{4} g x -40 b^{3} c \,e^{5} f x +1608 b^{2} c^{2} d^{2} e^{3} g x +336 b^{2} c^{2} d \,e^{4} f x -3120 b \,c^{3} d^{3} e^{2} g x -1056 b \,c^{3} d^{2} e^{3} f x +2000 c^{4} d^{4} e g x +1600 c^{4} d^{3} e^{2} f x +10 b^{4} d \,e^{4} g +35 b^{4} e^{5} f -94 b^{3} c \,d^{2} e^{3} g -320 b^{3} c d \,e^{4} f +348 b^{2} c^{2} d^{3} e^{2} g +1128 b^{2} c^{2} d^{2} e^{3} f -664 b \,c^{3} d^{4} e g -1856 b \,c^{3} d^{3} e^{2} f +400 c^{4} d^{5} g +1328 c^{4} d^{4} e f \right )}{315 \left (e x +d \right )^{4} \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 ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, 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 [B] time = 8.01, size = 949, normalized size = 2.64 \begin {gather*} \frac {\left (\frac {88\,c^2\,d\,g+96\,c^2\,e\,f-88\,b\,c\,e\,g}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {32\,c^3\,g\,\left (4\,b\,e-7\,c\,d\right )}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}-\frac {32\,c^4\,d\,g}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}-\frac {\left (\frac {4\,c\,g\,\left (5\,b\,e-8\,c\,d\right )}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}-\frac {8\,c^2\,d\,g}{63\,e\,\left (5\,b\,e^2-10\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {400\,c^3\,d\,g+384\,c^3\,e\,f-400\,b\,c^2\,e\,g}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}+\frac {16\,c^3\,d\,g}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {2\,b\,g}{9\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c\,d\,g}{9\,e\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^4}-\frac {\left (\frac {20\,c\,d\,g-20\,b\,e\,g+16\,c\,e\,f}{9\,e\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}+\frac {4\,c\,d\,g}{9\,e\,\left (7\,b\,e^2-14\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^4}+\frac {\left (\frac {2\,f}{9\,b\,e^2-18\,c\,d\,e}-\frac {2\,d\,g}{e\,\left (9\,b\,e^2-18\,c\,d\,e\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^5}+\frac {\left (\frac {16\,c^3\,d\,g}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}+\frac {16\,c^2\,g\,\left (2\,b\,e-5\,c\,d\right )}{315\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {736\,c^4\,d\,g+768\,c^4\,e\,f-736\,b\,c^3\,e\,g}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}-\frac {32\,c^4\,d\,g}{945\,e^2\,{\left (b\,e-2\,c\,d\right )}^5}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x} \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 \frac {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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