Optimal. Leaf size=346 \[ \frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65,
214} \begin {gather*} -\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (a+b x) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {1155 e^4 (a+b x) \sqrt {d+e x} (b d-a e)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{11/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (11 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (33 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^4 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (1155 e^3 \left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {1155 e^4 (b d-a e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {385 e^4 (a+b x) (d+e x)^{3/2}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1155 e^4 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.30, size = 300, normalized size = 0.87 \begin {gather*} \frac {e^4 (a+b x)^5 \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3465 a^5 e^5+1155 a^4 b e^4 (-4 d+11 e x)+231 a^3 b^2 e^3 \left (3 d^2-74 d e x+73 e^2 x^2\right )+99 a^2 b^3 e^2 \left (2 d^3+27 d^2 e x-232 d e^2 x^2+93 e^3 x^3\right )+11 a b^4 e \left (8 d^4+68 d^3 e x+345 d^2 e^2 x^2-1162 d e^3 x^3+128 e^4 x^4\right )+b^5 \left (48 d^5+328 d^4 e x+1030 d^3 e^2 x^2+2295 d^2 e^3 x^3-2048 d e^4 x^4-128 e^5 x^5\right )\right )}{e^4 (a+b x)^4}+3465 (-b d+a e)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{192 b^{13/2} \left ((a+b x)^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1470\) vs.
\(2(233)=466\).
time = 0.87, size = 1471, normalized size = 4.25
method | result | size |
risch | \(-\frac {2 e^{4} \left (-b e x +15 a e -16 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{6} \left (b x +a \right )}+\frac {\left (\frac {765 e^{5} \left (e x +d \right )^{\frac {7}{2}} a d}{32 b^{2} \left (b e x +a e \right )^{4}}-\frac {765 e^{6} \left (e x +d \right )^{\frac {7}{2}} a^{2}}{64 b^{3} \left (b e x +a e \right )^{4}}-\frac {5855 e^{7} \left (e x +d \right )^{\frac {5}{2}} a^{3}}{192 b^{4} \left (b e x +a e \right )^{4}}-\frac {5153 e^{8} \left (e x +d \right )^{\frac {3}{2}} a^{4}}{192 b^{5} \left (b e x +a e \right )^{4}}+\frac {5855 e^{6} \left (e x +d \right )^{\frac {5}{2}} a^{2} d}{64 b^{3} \left (b e x +a e \right )^{4}}-\frac {5855 e^{5} \left (e x +d \right )^{\frac {5}{2}} a \,d^{2}}{64 b^{2} \left (b e x +a e \right )^{4}}+\frac {5153 e^{7} \left (e x +d \right )^{\frac {3}{2}} a^{3} d}{48 b^{4} \left (b e x +a e \right )^{4}}-\frac {5153 e^{6} \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{2}}{32 b^{3} \left (b e x +a e \right )^{4}}+\frac {5153 e^{5} \left (e x +d \right )^{\frac {3}{2}} a \,d^{3}}{48 b^{2} \left (b e x +a e \right )^{4}}+\frac {2575 e^{8} \sqrt {e x +d}\, a^{4} d}{64 b^{5} \left (b e x +a e \right )^{4}}-\frac {2575 e^{7} \sqrt {e x +d}\, a^{3} d^{2}}{32 b^{4} \left (b e x +a e \right )^{4}}+\frac {2575 e^{6} \sqrt {e x +d}\, a^{2} d^{3}}{32 b^{3} \left (b e x +a e \right )^{4}}-\frac {2575 e^{5} \sqrt {e x +d}\, a \,d^{4}}{64 b^{2} \left (b e x +a e \right )^{4}}-\frac {1155 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a d}{32 b^{5} \sqrt {b \left (a e -b d \right )}}-\frac {765 e^{4} \left (e x +d \right )^{\frac {7}{2}} d^{2}}{64 b \left (b e x +a e \right )^{4}}+\frac {5855 e^{4} \left (e x +d \right )^{\frac {5}{2}} d^{3}}{192 b \left (b e x +a e \right )^{4}}-\frac {5153 e^{4} \left (e x +d \right )^{\frac {3}{2}} d^{4}}{192 b \left (b e x +a e \right )^{4}}-\frac {515 e^{9} \sqrt {e x +d}\, a^{5}}{64 b^{6} \left (b e x +a e \right )^{4}}+\frac {515 e^{4} \sqrt {e x +d}\, d^{5}}{64 b \left (b e x +a e \right )^{4}}+\frac {1155 e^{6} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2}}{64 b^{6} \sqrt {b \left (a e -b d \right )}}+\frac {1155 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{2}}{64 b^{4} \sqrt {b \left (a e -b d \right )}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(717\) |
default | \(\text {Expression too large to display}\) | \(1471\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.09, size = 893, normalized size = 2.58 \begin {gather*} \left [\frac {3465 \, {\left ({\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} e^{5} - {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d + 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (48 \, b^{5} d^{5} - {\left (128 \, b^{5} x^{5} - 1408 \, a b^{4} x^{4} - 9207 \, a^{2} b^{3} x^{3} - 16863 \, a^{3} b^{2} x^{2} - 12705 \, a^{4} b x - 3465 \, a^{5}\right )} e^{5} - 2 \, {\left (1024 \, b^{5} d x^{4} + 6391 \, a b^{4} d x^{3} + 11484 \, a^{2} b^{3} d x^{2} + 8547 \, a^{3} b^{2} d x + 2310 \, a^{4} b d\right )} e^{4} + 3 \, {\left (765 \, b^{5} d^{2} x^{3} + 1265 \, a b^{4} d^{2} x^{2} + 891 \, a^{2} b^{3} d^{2} x + 231 \, a^{3} b^{2} d^{2}\right )} e^{3} + 2 \, {\left (515 \, b^{5} d^{3} x^{2} + 374 \, a b^{4} d^{3} x + 99 \, a^{2} b^{3} d^{3}\right )} e^{2} + 8 \, {\left (41 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + d}}{384 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}, \frac {3465 \, {\left ({\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} e^{5} - {\left (b^{5} d x^{4} + 4 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 4 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (48 \, b^{5} d^{5} - {\left (128 \, b^{5} x^{5} - 1408 \, a b^{4} x^{4} - 9207 \, a^{2} b^{3} x^{3} - 16863 \, a^{3} b^{2} x^{2} - 12705 \, a^{4} b x - 3465 \, a^{5}\right )} e^{5} - 2 \, {\left (1024 \, b^{5} d x^{4} + 6391 \, a b^{4} d x^{3} + 11484 \, a^{2} b^{3} d x^{2} + 8547 \, a^{3} b^{2} d x + 2310 \, a^{4} b d\right )} e^{4} + 3 \, {\left (765 \, b^{5} d^{2} x^{3} + 1265 \, a b^{4} d^{2} x^{2} + 891 \, a^{2} b^{3} d^{2} x + 231 \, a^{3} b^{2} d^{2}\right )} e^{3} + 2 \, {\left (515 \, b^{5} d^{3} x^{2} + 374 \, a b^{4} d^{3} x + 99 \, a^{2} b^{3} d^{3}\right )} e^{2} + 8 \, {\left (41 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + d}}{192 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 500 vs.
\(2 (239) = 478\).
time = 0.79, size = 500, normalized size = 1.45 \begin {gather*} \frac {1155 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {2295 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d^{2} e^{4} - 5855 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{4} + 5153 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{4} - 1545 \, \sqrt {x e + d} b^{5} d^{5} e^{4} - 4590 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} d e^{5} + 17565 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{5} - 20612 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{5} + 7725 \, \sqrt {x e + d} a b^{4} d^{4} e^{5} + 2295 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{3} e^{6} - 17565 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{6} + 30918 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{6} - 15450 \, \sqrt {x e + d} a^{2} b^{3} d^{3} e^{6} + 5855 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{7} - 20612 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{7} + 15450 \, \sqrt {x e + d} a^{3} b^{2} d^{2} e^{7} + 5153 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b e^{8} - 7725 \, \sqrt {x e + d} a^{4} b d e^{8} + 1545 \, \sqrt {x e + d} a^{5} e^{9}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{10} e^{4} + 15 \, \sqrt {x e + d} b^{10} d e^{4} - 15 \, \sqrt {x e + d} a b^{9} e^{5}\right )}}{3 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{11/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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