Optimal. Leaf size=336 \[ -\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt {d+e x}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.61, 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} \[ \frac {16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{99 c^2 d^2 (d+e x)^{3/2}}+\frac {32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}{231 c^3 d^3 (d+e x)^{3/2}}+\frac {128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3}{1155 c^4 d^4 e \sqrt {d+e x}}-\frac {128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{3465 c^5 d^5 e (d+e x)^{3/2}}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rubi steps
\begin {align*} \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, 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 )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {(8 (c d f-a e g)) \int \frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{11 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 )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\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 )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (16 (c d f-a e g)^2\right ) \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{33 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 )^{3/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\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 )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\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 )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (64 (c d f-a e g)^3\right ) \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{231 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 )^{3/2}}{1155 c^4 d^4 e \sqrt {d+e x}}+\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 )^{3/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\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 )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\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 )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {\left (64 (c d f-a e g)^3 \left (5 f-\frac {3 d g}{e}-\frac {2 a e g}{c d}\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{1155 c^3 d^3}\\ &=\frac {128 (c d f-a e g)^3 \left (5 f-\frac {3 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 )^{3/2}}{3465 c^4 d^4 (d+e x)^{3/2}}+\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 )^{3/2}}{1155 c^4 d^4 e \sqrt {d+e x}}+\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 )^{3/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\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 )^{3/2}}{99 c^2 d^2 (d+e x)^{3/2}}+\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 )^{3/2}}{11 c d (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 195, normalized size = 0.58 \[ \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (11 f+3 g x)+48 a^2 c^2 d^2 e^2 g^2 \left (33 f^2+22 f g x+5 g^2 x^2\right )-8 a c^3 d^3 e g \left (231 f^3+297 f^2 g x+165 f g^2 x^2+35 g^3 x^3\right )+c^4 d^4 \left (1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 375, normalized size = 1.12 \[ \frac {2 \, {\left (315 \, c^{5} d^{5} g^{4} x^{5} + 1155 \, a c^{4} d^{4} e f^{4} - 1848 \, a^{2} c^{3} d^{3} e^{2} f^{3} g + 1584 \, a^{3} c^{2} d^{2} e^{3} f^{2} g^{2} - 704 \, a^{4} c d e^{4} f g^{3} + 128 \, a^{5} e^{5} g^{4} + 35 \, {\left (44 \, c^{5} d^{5} f g^{3} + a c^{4} d^{4} e g^{4}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{5} f^{2} g^{2} + 22 \, a c^{4} d^{4} e f g^{3} - 4 \, a^{2} c^{3} d^{3} e^{2} g^{4}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{5} f^{3} g + 99 \, a c^{4} d^{4} e f^{2} g^{2} - 44 \, a^{2} c^{3} d^{3} e^{2} f g^{3} + 8 \, a^{3} c^{2} d^{2} e^{3} g^{4}\right )} x^{2} + {\left (1155 \, c^{5} d^{5} f^{4} + 924 \, a c^{4} d^{4} e f^{3} g - 792 \, a^{2} c^{3} d^{3} e^{2} f^{2} g^{2} + 352 \, a^{3} c^{2} d^{2} e^{3} f g^{3} - 64 \, a^{4} c d e^{4} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3465 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
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 \[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{4}}{\sqrt {e x + d}}\,{d x} \]
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 \[ \frac {2 \left (c d x +a e \right ) \left (315 g^{4} x^{4} c^{4} d^{4}-280 a \,c^{3} d^{3} e \,g^{4} x^{3}+1540 c^{4} d^{4} f \,g^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-1320 a \,c^{3} d^{3} e f \,g^{3} x^{2}+2970 c^{4} d^{4} f^{2} g^{2} x^{2}-192 a^{3} c d \,e^{3} g^{4} x +1056 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -2376 a \,c^{3} d^{3} e \,f^{2} g^{2} x +2772 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-704 a^{3} c d \,e^{3} f \,g^{3}+1584 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-1848 a \,c^{3} d^{3} e \,f^{3} g +1155 f^{4} c^{4} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3465 \sqrt {e x +d}\, c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 320, normalized size = 0.95 \[ \frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}} f^{4}}{3 \, c d} + \frac {8 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{3} g}{15 \, c^{2} d^{2}} + \frac {4 \, {\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{35 \, c^{3} d^{3}} + \frac {8 \, {\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f g^{3}}{315 \, c^{4} d^{4}} + \frac {2 \, {\left (315 \, c^{5} d^{5} x^{5} + 35 \, a c^{4} d^{4} e x^{4} - 40 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 48 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 64 \, a^{4} c d e^{4} x + 128 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} g^{4}}{3465 \, c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 347, normalized size = 1.03 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^4\,x^5}{11}+\frac {256\,a^5\,e^5\,g^4-1408\,a^4\,c\,d\,e^4\,f\,g^3+3168\,a^3\,c^2\,d^2\,e^3\,f^2\,g^2-3696\,a^2\,c^3\,d^3\,e^2\,f^3\,g+2310\,a\,c^4\,d^4\,e\,f^4}{3465\,c^5\,d^5}+\frac {x\,\left (-128\,a^4\,c\,d\,e^4\,g^4+704\,a^3\,c^2\,d^2\,e^3\,f\,g^3-1584\,a^2\,c^3\,d^3\,e^2\,f^2\,g^2+1848\,a\,c^4\,d^4\,e\,f^3\,g+2310\,c^5\,d^5\,f^4\right )}{3465\,c^5\,d^5}+\frac {4\,g\,x^2\,\left (8\,a^3\,e^3\,g^3-44\,a^2\,c\,d\,e^2\,f\,g^2+99\,a\,c^2\,d^2\,e\,f^2\,g+462\,c^3\,d^3\,f^3\right )}{1155\,c^3\,d^3}+\frac {4\,g^2\,x^3\,\left (-4\,a^2\,e^2\,g^2+22\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,c^2\,d^2}+\frac {2\,g^3\,x^4\,\left (a\,e\,g+44\,c\,d\,f\right )}{99\,c\,d}\right )}{\sqrt {d+e\,x}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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