Optimal. Leaf size=164 \[ \frac {2 (A b-a B) (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {2 (A b-a B) (d+e x)^{5/2}}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e}-\frac {2 (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {81, 52, 65, 214}
\begin {gather*} -\frac {2 (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}}+\frac {2 \sqrt {d+e x} (A b-a B) (b d-a e)^2}{b^4}+\frac {2 (d+e x)^{3/2} (A b-a B) (b d-a e)}{3 b^3}+\frac {2 (d+e x)^{5/2} (A b-a B)}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 214
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{5/2}}{a+b x} \, dx &=\frac {2 B (d+e x)^{7/2}}{7 b e}+\frac {\left (2 \left (\frac {7 A b e}{2}-\frac {7 a B e}{2}\right )\right ) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{7 b e}\\ &=\frac {2 (A b-a B) (d+e x)^{5/2}}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e}+\frac {((A b-a B) (b d-a e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{b^2}\\ &=\frac {2 (A b-a B) (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {2 (A b-a B) (d+e x)^{5/2}}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e}+\frac {\left ((A b-a B) (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{b^3}\\ &=\frac {2 (A b-a B) (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {2 (A b-a B) (d+e x)^{5/2}}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e}+\frac {\left ((A b-a B) (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b^4}\\ &=\frac {2 (A b-a B) (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {2 (A b-a B) (d+e x)^{5/2}}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e}+\frac {\left (2 (A b-a B) (b d-a e)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^4 e}\\ &=\frac {2 (A b-a B) (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {2 (A b-a B) (d+e x)^{5/2}}{5 b^2}+\frac {2 B (d+e x)^{7/2}}{7 b e}-\frac {2 (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 185, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {d+e x} \left (-105 a^3 B e^3+35 a^2 b e^2 (7 B d+3 A e+B e x)-7 a b^2 e \left (5 A e (7 d+e x)+B \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )+b^3 \left (15 B (d+e x)^3+7 A e \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )\right )}{105 b^4 e}-\frac {2 (A b-a B) (-b d+a e)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{9/2}} \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. \(345\) vs.
\(2(140)=280\).
time = 0.09, size = 346, normalized size = 2.11
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {b^{3} B \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{2} b \,e^{3} \sqrt {e x +d}-2 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+A \,b^{3} d^{2} e \sqrt {e x +d}-B \,a^{3} e^{3} \sqrt {e x +d}+2 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-B a \,b^{2} d^{2} e \sqrt {e x +d}\right )}{b^{4}}-\frac {2 e \left (A \,a^{3} b \,e^{3}-3 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e -A \,b^{4} d^{3}-B \,a^{4} e^{3}+3 B \,a^{3} b d \,e^{2}-3 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{4} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(346\) |
default | \(\frac {\frac {2 \left (\frac {b^{3} B \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+A \,a^{2} b \,e^{3} \sqrt {e x +d}-2 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+A \,b^{3} d^{2} e \sqrt {e x +d}-B \,a^{3} e^{3} \sqrt {e x +d}+2 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-B a \,b^{2} d^{2} e \sqrt {e x +d}\right )}{b^{4}}-\frac {2 e \left (A \,a^{3} b \,e^{3}-3 A \,a^{2} b^{2} d \,e^{2}+3 A a \,b^{3} d^{2} e -A \,b^{4} d^{3}-B \,a^{4} e^{3}+3 B \,a^{3} b d \,e^{2}-3 B \,a^{2} b^{2} d^{2} e +B a \,b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{b^{4} \sqrt {\left (a e -b d \right ) b}}}{e}\) | \(346\) |
risch | \(\frac {2 \left (15 b^{3} B \,x^{3} e^{3}+21 A \,b^{3} e^{3} x^{2}-21 B a \,b^{2} e^{3} x^{2}+45 B \,b^{3} d \,e^{2} x^{2}-35 A a \,b^{2} e^{3} x +77 A \,b^{3} d \,e^{2} x +35 B \,a^{2} b \,e^{3} x -77 B a \,b^{2} d \,e^{2} x +45 B \,b^{3} d^{2} e x +105 A \,a^{2} b \,e^{3}-245 A a \,b^{2} d \,e^{2}+161 A \,b^{3} d^{2} e -105 B \,a^{3} e^{3}+245 B \,a^{2} b d \,e^{2}-161 B a \,b^{2} d^{2} e +15 b^{3} B \,d^{3}\right ) \sqrt {e x +d}}{105 e \,b^{4}}-\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,a^{3} e^{3}}{b^{3} \sqrt {\left (a e -b d \right ) b}}+\frac {6 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,a^{2} d \,e^{2}}{b^{2} \sqrt {\left (a e -b d \right ) b}}-\frac {6 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A a \,d^{2} e}{b \sqrt {\left (a e -b d \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) A \,d^{3}}{\sqrt {\left (a e -b d \right ) b}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{4} e^{3}}{b^{4} \sqrt {\left (a e -b d \right ) b}}-\frac {6 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{3} d \,e^{2}}{b^{3} \sqrt {\left (a e -b d \right ) b}}+\frac {6 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B \,a^{2} d^{2} e}{b^{2} \sqrt {\left (a e -b d \right ) b}}-\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) B a \,d^{3}}{b \sqrt {\left (a e -b d \right ) b}}\) | \(557\) |
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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Fricas [A]
time = 1.00, size = 572, normalized size = 3.49 \begin {gather*} \left [\frac {{\left (105 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} + {\left (B a^{3} - A a^{2} b\right )} e^{3}\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 (15 \, B b^{3} d^{3} + {\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} e^{3} + {\left (45 \, B b^{3} d x^{2} - 77 \, {\left (B a b^{2} - A b^{3}\right )} d x + 245 \, {\left (B a^{2} b - A a b^{2}\right )} d\right )} e^{2} + {\left (45 \, B b^{3} d^{2} x - 161 \, {\left (B a b^{2} - A b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{105 \, b^{4}}, \frac {2 \, {\left (105 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e^{2} + {\left (B a^{3} - A a^{2} b\right )} e^{3}\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 (15 \, B b^{3} d^{3} + {\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} e^{3} + {\left (45 \, B b^{3} d x^{2} - 77 \, {\left (B a b^{2} - A b^{3}\right )} d x + 245 \, {\left (B a^{2} b - A a b^{2}\right )} d\right )} e^{2} + {\left (45 \, B b^{3} d^{2} x - 161 \, {\left (B a b^{2} - A b^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}\right )} e^{\left (-1\right )}}{105 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 35.80, size = 221, normalized size = 1.35 \begin {gather*} \frac {2 B \left (d + e x\right )^{\frac {7}{2}}}{7 b e} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A b - 2 B a\right )}{5 b^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{3 b^{3}} + \frac {\sqrt {d + e x} \left (2 A a^{2} b e^{2} - 4 A a b^{2} d e + 2 A b^{3} d^{2} - 2 B a^{3} e^{2} + 4 B a^{2} b d e - 2 B a b^{2} d^{2}\right )}{b^{4}} + \frac {2 \left (- A b + B a\right ) \left (a e - b d\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{5} \sqrt {\frac {a e - b d}{b}}} \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. 371 vs.
\(2 (148) = 296\).
time = 0.86, size = 371, normalized size = 2.26 \begin {gather*} -\frac {2 \, {\left (B a b^{3} d^{3} - A b^{4} d^{3} - 3 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 3 \, B a^{3} b d e^{2} - 3 \, A a^{2} b^{2} d e^{2} - B a^{4} e^{3} + A a^{3} b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{6} e^{6} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{5} e^{7} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{6} e^{7} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{5} d e^{7} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{6} d e^{7} - 105 \, \sqrt {x e + d} B a b^{5} d^{2} e^{7} + 105 \, \sqrt {x e + d} A b^{6} d^{2} e^{7} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{4} e^{8} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{5} e^{8} + 210 \, \sqrt {x e + d} B a^{2} b^{4} d e^{8} - 210 \, \sqrt {x e + d} A a b^{5} d e^{8} - 105 \, \sqrt {x e + d} B a^{3} b^{3} e^{9} + 105 \, \sqrt {x e + d} A a^{2} b^{4} e^{9}\right )} e^{\left (-7\right )}}{105 \, b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 330, normalized size = 2.01 \begin {gather*} \left (\frac {2\,A\,e-2\,B\,d}{5\,b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{5\,b^2\,e^2}\right )\,{\left (d+e\,x\right )}^{5/2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{B\,a^4\,e^3-3\,B\,a^3\,b\,d\,e^2-A\,a^3\,b\,e^3+3\,B\,a^2\,b^2\,d^2\,e+3\,A\,a^2\,b^2\,d\,e^2-B\,a\,b^3\,d^3-3\,A\,a\,b^3\,d^2\,e+A\,b^4\,d^3}\right )\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{b^{9/2}}+\frac {2\,B\,{\left (d+e\,x\right )}^{7/2}}{7\,b\,e}+\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,{\left (a\,e^2-b\,d\,e\right )}^2\,\sqrt {d+e\,x}}{b^2\,e^2}-\frac {\left (\frac {2\,A\,e-2\,B\,d}{b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{b^2\,e^2}\right )\,\left (a\,e^2-b\,d\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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