Optimal. Leaf size=295 \[ \frac {256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662}
\begin {gather*} \frac {256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac {16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 662
Rule 670
Rubi steps
\begin {align*} \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{11 d}\\ &=\frac {16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \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}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{33 d^2}\\ &=\frac {32 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \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}+\frac {16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \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}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (64 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{231 d^3}\\ &=\frac {128 \left (c d^2-a e^2\right )^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 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \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}+\frac {16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \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}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (128 \left (d^2-\frac {a e^2}{c}\right )^4\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 d^4}\\ &=\frac {256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^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 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \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}+\frac {16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \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}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 187, normalized size = 0.63 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2} \left (128 a^4 e^8-64 a^3 c d e^6 (11 d+3 e x)+48 a^2 c^2 d^2 e^4 \left (33 d^2+22 d e x+5 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (231 d^3+297 d^2 e x+165 d e^2 x^2+35 e^3 x^3\right )+c^4 d^4 \left (1155 d^4+2772 d^3 e x+2970 d^2 e^2 x^2+1540 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.73, size = 233, normalized size = 0.79
method | result | size |
default | \(\frac {2 \left (c d x +a e \right ) \left (315 e^{4} x^{4} c^{4} d^{4}-280 a \,c^{3} d^{3} e^{5} x^{3}+1540 c^{4} d^{5} e^{3} x^{3}+240 e^{6} a^{2} x^{2} c^{2} d^{2}-1320 e^{4} a \,x^{2} c^{3} d^{4}+2970 e^{2} x^{2} c^{4} d^{6}-192 a^{3} c d \,e^{7} x +1056 a^{2} c^{2} d^{3} e^{5} x -2376 a \,c^{3} d^{5} e^{3} x +2772 c^{4} d^{7} e x +128 a^{4} e^{8}-704 a^{3} c \,d^{2} e^{6}+1584 a^{2} c^{2} d^{4} e^{4}-1848 a \,c^{3} d^{6} e^{2}+1155 c^{4} d^{8}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{3465 c^{5} d^{5} \sqrt {e x +d}}\) | \(233\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (315 e^{4} x^{4} c^{4} d^{4}-280 a \,c^{3} d^{3} e^{5} x^{3}+1540 c^{4} d^{5} e^{3} x^{3}+240 e^{6} a^{2} x^{2} c^{2} d^{2}-1320 e^{4} a \,x^{2} c^{3} d^{4}+2970 e^{2} x^{2} c^{4} d^{6}-192 a^{3} c d \,e^{7} x +1056 a^{2} c^{2} d^{3} e^{5} x -2376 a \,c^{3} d^{5} e^{3} x +2772 c^{4} d^{7} e x +128 a^{4} e^{8}-704 a^{3} c \,d^{2} e^{6}+1584 a^{2} c^{2} d^{4} e^{4}-1848 a \,c^{3} d^{6} e^{2}+1155 c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3465 c^{5} d^{5} \sqrt {e x +d}}\) | \(243\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 284, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (315 \, c^{5} d^{5} x^{5} e^{4} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \, {\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt {c d x + a e} {\left (x e + d\right )}}{3465 \, {\left (c^{5} d^{5} x e + c^{5} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.47, size = 313, normalized size = 1.06 \begin {gather*} \frac {2 \, {\left (1155 \, c^{5} d^{9} x - 64 \, a^{4} c d x e^{8} + 128 \, a^{5} e^{9} + 16 \, {\left (3 \, a^{3} c^{2} d^{2} x^{2} - 44 \, a^{4} c d^{2}\right )} e^{7} - 8 \, {\left (5 \, a^{2} c^{3} d^{3} x^{3} - 44 \, a^{3} c^{2} d^{3} x\right )} e^{6} + {\left (35 \, a c^{4} d^{4} x^{4} - 264 \, a^{2} c^{3} d^{4} x^{2} + 1584 \, a^{3} c^{2} d^{4}\right )} e^{5} + {\left (315 \, c^{5} d^{5} x^{5} + 220 \, a c^{4} d^{5} x^{3} - 792 \, a^{2} c^{3} d^{5} x\right )} e^{4} + 22 \, {\left (70 \, c^{5} d^{6} x^{4} + 27 \, a c^{4} d^{6} x^{2} - 84 \, a^{2} c^{3} d^{6}\right )} e^{3} + 66 \, {\left (45 \, c^{5} d^{7} x^{3} + 14 \, a c^{4} d^{7} x\right )} e^{2} + 231 \, {\left (12 \, c^{5} d^{8} x^{2} + 5 \, a c^{4} d^{8}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3465 \, {\left (c^{5} d^{5} x e + c^{5} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1078 vs.
\(2 (271) = 542\).
time = 1.71, size = 1078, normalized size = 3.65 \begin {gather*} \frac {2}{3465} \, {\left (1155 \, d^{4} {\left (\frac {{\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{c d} + \frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} - \sqrt {-c d^{2} e + a e^{3}} a e^{2}}{c d}\right )} e^{\left (-1\right )} - 924 \, d^{3} {\left (\frac {{\left (5 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )}}{c^{2} d^{2}} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}}\right )} e^{\left (-1\right )} + 198 \, d^{2} {\left (\frac {{\left (15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}\right )} e^{\left (-2\right )}}{c^{3} d^{3}} + \frac {{\left (35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )}}{c^{3} d^{3}}\right )} e - 44 \, d {\left (\frac {{\left (35 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{8} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{6} e^{2} - 6 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{4} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d^{2} e^{6} - 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} e^{8}\right )} e^{\left (-3\right )}}{c^{4} d^{4}} + \frac {{\left (105 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}\right )} e^{\left (-7\right )}}{c^{4} d^{4}}\right )} e^{2} + {\left (\frac {{\left (315 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{10} - 35 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{8} e^{2} - 40 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{3} d^{6} e^{4} - 48 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c^{2} d^{4} e^{6} - 64 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} c d^{2} e^{8} - 128 \, \sqrt {-c d^{2} e + a e^{3}} a^{5} e^{10}\right )} e^{\left (-4\right )}}{c^{5} d^{5}} + \frac {{\left (1155 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{4} e^{12} - 2772 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{3} e^{9} + 2970 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a^{2} e^{6} - 1540 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} a e^{3} + 315 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {11}{2}}\right )} e^{\left (-9\right )}}{c^{5} d^{5}}\right )} e^{3}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.21, size = 346, normalized size = 1.17 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e^3\,x^5\,\sqrt {d+e\,x}}{11}+\frac {4\,x^2\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-44\,a^2\,c\,d^2\,e^4+99\,a\,c^2\,d^4\,e^2+462\,c^3\,d^6\right )}{1155\,c^3\,d^3}+\frac {\sqrt {d+e\,x}\,\left (256\,a^5\,e^9-1408\,a^4\,c\,d^2\,e^7+3168\,a^3\,c^2\,d^4\,e^5-3696\,a^2\,c^3\,d^6\,e^3+2310\,a\,c^4\,d^8\,e\right )}{3465\,c^5\,d^5\,e}+\frac {2\,e^2\,x^4\,\left (44\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{99\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,a^4\,c\,d\,e^8+704\,a^3\,c^2\,d^3\,e^6-1584\,a^2\,c^3\,d^5\,e^4+1848\,a\,c^4\,d^7\,e^2+2310\,c^5\,d^9\right )}{3465\,c^5\,d^5\,e}+\frac {4\,e\,x^3\,\sqrt {d+e\,x}\,\left (-4\,a^2\,e^4+22\,a\,c\,d^2\,e^2+297\,c^2\,d^4\right )}{693\,c^2\,d^2}\right )}{x+\frac {d}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________