Optimal. Leaf size=336 \[ -\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \]
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Rubi [A] time = 0.62, antiderivative size = 336, 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 = {870, 794, 648} \begin {gather*} \frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}-\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rubi steps
\begin {align*} \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {(8 (c d f-a e g)) \int \frac {(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{15 c d}\\ &=\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {\left (16 (c d f-a e g)^2\right ) \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{65 c^2 d^2}\\ &=\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {\left (64 (c d f-a e g)^3\right ) \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{715 c^3 d^3}\\ &=\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac {\left (64 (c d f-a e g)^3 \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{6435 c^3 d^3}\\ &=\frac {128 (c d f-a e g)^3 \left (9 f-\frac {7 d g}{e}-\frac {2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{45045 c^4 d^4 (d+e x)^{7/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 205, normalized size = 0.61 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (15 f+7 g x)+48 a^2 c^2 d^2 e^2 g^2 \left (65 f^2+70 f g x+21 g^2 x^2\right )-8 a c^3 d^3 e g \left (715 f^3+1365 f^2 g x+945 f g^2 x^2+231 g^3 x^3\right )+c^4 d^4 \left (6435 f^4+20020 f^3 g x+24570 f^2 g^2 x^2+13860 f g^3 x^3+3003 g^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 567, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (3003 \, c^{7} d^{7} g^{4} x^{7} + 6435 \, a^{3} c^{4} d^{4} e^{3} f^{4} - 5720 \, a^{4} c^{3} d^{3} e^{4} f^{3} g + 3120 \, a^{5} c^{2} d^{2} e^{5} f^{2} g^{2} - 960 \, a^{6} c d e^{6} f g^{3} + 128 \, a^{7} e^{7} g^{4} + 231 \, {\left (60 \, c^{7} d^{7} f g^{3} + 31 \, a c^{6} d^{6} e g^{4}\right )} x^{6} + 63 \, {\left (390 \, c^{7} d^{7} f^{2} g^{2} + 540 \, a c^{6} d^{6} e f g^{3} + 71 \, a^{2} c^{5} d^{5} e^{2} g^{4}\right )} x^{5} + 35 \, {\left (572 \, c^{7} d^{7} f^{3} g + 1794 \, a c^{6} d^{6} e f^{2} g^{2} + 636 \, a^{2} c^{5} d^{5} e^{2} f g^{3} + a^{3} c^{4} d^{4} e^{3} g^{4}\right )} x^{4} + 5 \, {\left (1287 \, c^{7} d^{7} f^{4} + 10868 \, a c^{6} d^{6} e f^{3} g + 8814 \, a^{2} c^{5} d^{5} e^{2} f^{2} g^{2} + 60 \, a^{3} c^{4} d^{4} e^{3} f g^{3} - 8 \, a^{4} c^{3} d^{3} e^{4} g^{4}\right )} x^{3} + 3 \, {\left (6435 \, a c^{6} d^{6} e f^{4} + 14300 \, a^{2} c^{5} d^{5} e^{2} f^{3} g + 390 \, a^{3} c^{4} d^{4} e^{3} f^{2} g^{2} - 120 \, a^{4} c^{3} d^{3} e^{4} f g^{3} + 16 \, a^{5} c^{2} d^{2} e^{5} g^{4}\right )} x^{2} + {\left (19305 \, a^{2} c^{5} d^{5} e^{2} f^{4} + 2860 \, a^{3} c^{4} d^{4} e^{3} f^{3} g - 1560 \, a^{4} c^{3} d^{3} e^{4} f^{2} g^{2} + 480 \, a^{5} c^{2} d^{2} e^{5} f g^{3} - 64 \, a^{6} c d e^{6} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{45045 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 283, normalized size = 0.84 \begin {gather*} \frac {2 \left (c d x +a e \right ) \left (3003 g^{4} x^{4} c^{4} d^{4}-1848 a \,c^{3} d^{3} e \,g^{4} x^{3}+13860 c^{4} d^{4} f \,g^{3} x^{3}+1008 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-7560 a \,c^{3} d^{3} e f \,g^{3} x^{2}+24570 c^{4} d^{4} f^{2} g^{2} x^{2}-448 a^{3} c d \,e^{3} g^{4} x +3360 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -10920 a \,c^{3} d^{3} e \,f^{2} g^{2} x +20020 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-960 a^{3} c d \,e^{3} f \,g^{3}+3120 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-5720 a \,c^{3} d^{3} e \,f^{3} g +6435 f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{45045 \left (e x +d \right )^{\frac {5}{2}} c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 498, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{4}}{7 \, c d} + \frac {8 \, {\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f^{3} g}{63 \, c^{2} d^{2}} + \frac {4 \, {\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{231 \, c^{3} d^{3}} + \frac {8 \, {\left (231 \, c^{6} d^{6} x^{6} + 567 \, a c^{5} d^{5} e x^{5} + 371 \, a^{2} c^{4} d^{4} e^{2} x^{4} + 5 \, a^{3} c^{3} d^{3} e^{3} x^{3} - 6 \, a^{4} c^{2} d^{2} e^{4} x^{2} + 8 \, a^{5} c d e^{5} x - 16 \, a^{6} e^{6}\right )} \sqrt {c d x + a e} f g^{3}}{3003 \, c^{4} d^{4}} + \frac {2 \, {\left (3003 \, c^{7} d^{7} x^{7} + 7161 \, a c^{6} d^{6} e x^{6} + 4473 \, a^{2} c^{5} d^{5} e^{2} x^{5} + 35 \, a^{3} c^{4} d^{4} e^{3} x^{4} - 40 \, a^{4} c^{3} d^{3} e^{4} x^{3} + 48 \, a^{5} c^{2} d^{2} e^{5} x^{2} - 64 \, a^{6} c d e^{6} x + 128 \, a^{7} e^{7}\right )} \sqrt {c d x + a e} g^{4}}{45045 \, c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.09, size = 523, normalized size = 1.56 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^2\,x^5\,\left (71\,a^2\,e^2\,g^2+540\,a\,c\,d\,e\,f\,g+390\,c^2\,d^2\,f^2\right )}{715}+\frac {256\,a^7\,e^7\,g^4-1920\,a^6\,c\,d\,e^6\,f\,g^3+6240\,a^5\,c^2\,d^2\,e^5\,f^2\,g^2-11440\,a^4\,c^3\,d^3\,e^4\,f^3\,g+12870\,a^3\,c^4\,d^4\,e^3\,f^4}{45045\,c^5\,d^5}+\frac {x^3\,\left (-80\,a^4\,c^3\,d^3\,e^4\,g^4+600\,a^3\,c^4\,d^4\,e^3\,f\,g^3+88140\,a^2\,c^5\,d^5\,e^2\,f^2\,g^2+108680\,a\,c^6\,d^6\,e\,f^3\,g+12870\,c^7\,d^7\,f^4\right )}{45045\,c^5\,d^5}+\frac {2\,c^2\,d^2\,g^4\,x^7}{15}+\frac {2\,c\,d\,g^3\,x^6\,\left (31\,a\,e\,g+60\,c\,d\,f\right )}{195}+\frac {2\,g\,x^4\,\left (a^3\,e^3\,g^3+636\,a^2\,c\,d\,e^2\,f\,g^2+1794\,a\,c^2\,d^2\,e\,f^2\,g+572\,c^3\,d^3\,f^3\right )}{1287\,c\,d}+\frac {2\,a^2\,e^2\,x\,\left (-64\,a^4\,e^4\,g^4+480\,a^3\,c\,d\,e^3\,f\,g^3-1560\,a^2\,c^2\,d^2\,e^2\,f^2\,g^2+2860\,a\,c^3\,d^3\,e\,f^3\,g+19305\,c^4\,d^4\,f^4\right )}{45045\,c^4\,d^4}+\frac {2\,a\,e\,x^2\,\left (16\,a^4\,e^4\,g^4-120\,a^3\,c\,d\,e^3\,f\,g^3+390\,a^2\,c^2\,d^2\,e^2\,f^2\,g^2+14300\,a\,c^3\,d^3\,e\,f^3\,g+6435\,c^4\,d^4\,f^4\right )}{15015\,c^3\,d^3}\right )}{\sqrt {d+e\,x}} \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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