Optimal. Leaf size=198 \[ -\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {2 \sqrt {d+e x} (A b-a B) (b d-a e)^3}{b^5}+\frac {2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac {2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac {2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e} \]
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Rubi [A] time = 0.21, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {80, 50, 63, 208} \begin {gather*} \frac {2 (d+e x)^{7/2} (A b-a B)}{7 b^2}+\frac {2 (d+e x)^{5/2} (A b-a B) (b d-a e)}{5 b^3}+\frac {2 (d+e x)^{3/2} (A b-a B) (b d-a e)^2}{3 b^4}+\frac {2 \sqrt {d+e x} (A b-a B) (b d-a e)^3}{b^5}-\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}+\frac {2 B (d+e x)^{9/2}}{9 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{a+b x} \, dx &=\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 \left (\frac {9 A b e}{2}-\frac {9 a B e}{2}\right )\right ) \int \frac {(d+e x)^{7/2}}{a+b x} \, dx}{9 b e}\\ &=\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {((A b-a B) (b d-a e)) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{b^2}\\ &=\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^2\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{b^3}\\ &=\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{b^4}\\ &=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left ((A b-a B) (b d-a e)^4\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b^5}\\ &=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}+\frac {\left (2 (A b-a B) (b d-a e)^4\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^5 e}\\ &=\frac {2 (A b-a B) (b d-a e)^3 \sqrt {d+e x}}{b^5}+\frac {2 (A b-a B) (b d-a e)^2 (d+e x)^{3/2}}{3 b^4}+\frac {2 (A b-a B) (b d-a e) (d+e x)^{5/2}}{5 b^3}+\frac {2 (A b-a B) (d+e x)^{7/2}}{7 b^2}+\frac {2 B (d+e x)^{9/2}}{9 b e}-\frac {2 (A b-a B) (b d-a e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 165, normalized size = 0.83 \begin {gather*} \frac {2 \left (\frac {3 e (A b-a B) \left (7 (b d-a e) \left (5 (b d-a e) \left (\sqrt {b} \sqrt {d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )+3 b^{5/2} (d+e x)^{5/2}\right )+15 b^{7/2} (d+e x)^{7/2}\right )}{35 b^{9/2}}+B (d+e x)^{9/2}\right )}{9 b e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.30, size = 485, normalized size = 2.45 \begin {gather*} \frac {2 \left (315 a^4 B e^4 \sqrt {d+e x}-315 a^3 A b e^4 \sqrt {d+e x}-105 a^3 b B e^3 (d+e x)^{3/2}-945 a^3 b B d e^3 \sqrt {d+e x}+105 a^2 A b^2 e^3 (d+e x)^{3/2}+945 a^2 A b^2 d e^3 \sqrt {d+e x}+945 a^2 b^2 B d^2 e^2 \sqrt {d+e x}+63 a^2 b^2 B e^2 (d+e x)^{5/2}+210 a^2 b^2 B d e^2 (d+e x)^{3/2}-945 a A b^3 d^2 e^2 \sqrt {d+e x}-63 a A b^3 e^2 (d+e x)^{5/2}-210 a A b^3 d e^2 (d+e x)^{3/2}-315 a b^3 B d^3 e \sqrt {d+e x}-105 a b^3 B d^2 e (d+e x)^{3/2}-45 a b^3 B e (d+e x)^{7/2}-63 a b^3 B d e (d+e x)^{5/2}+315 A b^4 d^3 e \sqrt {d+e x}+105 A b^4 d^2 e (d+e x)^{3/2}+45 A b^4 e (d+e x)^{7/2}+63 A b^4 d e (d+e x)^{5/2}+35 b^4 B (d+e x)^{9/2}\right )}{315 b^5 e}-\frac {2 (A b-a B) (a e-b d)^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.41, size = 865, normalized size = 4.37 \begin {gather*} \left [-\frac {315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (35 \, B b^{4} e^{4} x^{4} + 35 \, B b^{4} d^{4} - 528 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 1218 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 1050 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + 315 \, {\left (B a^{4} - A a^{3} b\right )} e^{4} + 5 \, {\left (28 \, B b^{4} d e^{3} - 9 \, {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (70 \, B b^{4} d^{2} e^{2} - 66 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + 21 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{2} + {\left (140 \, B b^{4} d^{3} e - 366 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 336 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, b^{5} e}, \frac {2 \, {\left (315 \, {\left ({\left (B a b^{3} - A b^{4}\right )} d^{3} e - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} - {\left (B a^{4} - A a^{3} b\right )} e^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (35 \, B b^{4} e^{4} x^{4} + 35 \, B b^{4} d^{4} - 528 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} e + 1218 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{2} - 1050 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{3} + 315 \, {\left (B a^{4} - A a^{3} b\right )} e^{4} + 5 \, {\left (28 \, B b^{4} d e^{3} - 9 \, {\left (B a b^{3} - A b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (70 \, B b^{4} d^{2} e^{2} - 66 \, {\left (B a b^{3} - A b^{4}\right )} d e^{3} + 21 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{4}\right )} x^{2} + {\left (140 \, B b^{4} d^{3} e - 366 \, {\left (B a b^{3} - A b^{4}\right )} d^{2} e^{2} + 336 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d e^{3} - 105 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} e^{4}\right )} x\right )} \sqrt {e x + d}\right )}}{315 \, b^{5} e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.41, size = 558, normalized size = 2.82 \begin {gather*} -\frac {2 \, {\left (B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}\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^{5}} + \frac {2 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{8} e^{8} - 45 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{7} e^{9} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{8} e^{9} - 63 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{7} d e^{9} + 63 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{8} d e^{9} - 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{7} d^{2} e^{9} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{8} d^{2} e^{9} - 315 \, \sqrt {x e + d} B a b^{7} d^{3} e^{9} + 315 \, \sqrt {x e + d} A b^{8} d^{3} e^{9} + 63 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{6} e^{10} - 63 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{7} e^{10} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{6} d e^{10} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{7} d e^{10} + 945 \, \sqrt {x e + d} B a^{2} b^{6} d^{2} e^{10} - 945 \, \sqrt {x e + d} A a b^{7} d^{2} e^{10} - 105 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{5} e^{11} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{6} e^{11} - 945 \, \sqrt {x e + d} B a^{3} b^{5} d e^{11} + 945 \, \sqrt {x e + d} A a^{2} b^{6} d e^{11} + 315 \, \sqrt {x e + d} B a^{4} b^{4} e^{12} - 315 \, \sqrt {x e + d} A a^{3} b^{5} e^{12}\right )} e^{\left (-9\right )}}{315 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 820, normalized size = 4.14 \begin {gather*} \frac {2 A \,a^{4} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{4}}-\frac {8 A \,a^{3} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {12 A \,a^{2} d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {8 A a \,d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}+\frac {2 A \,d^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}-\frac {2 B \,a^{5} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{5}}+\frac {8 B \,a^{4} d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{4}}-\frac {12 B \,a^{3} d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {8 B \,a^{2} d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{2}}-\frac {2 B a \,d^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}-\frac {2 \sqrt {e x +d}\, A \,a^{3} e^{3}}{b^{4}}+\frac {6 \sqrt {e x +d}\, A \,a^{2} d \,e^{2}}{b^{3}}-\frac {6 \sqrt {e x +d}\, A a \,d^{2} e}{b^{2}}+\frac {2 \sqrt {e x +d}\, A \,d^{3}}{b}+\frac {2 \sqrt {e x +d}\, B \,a^{4} e^{3}}{b^{5}}-\frac {6 \sqrt {e x +d}\, B \,a^{3} d \,e^{2}}{b^{4}}+\frac {6 \sqrt {e x +d}\, B \,a^{2} d^{2} e}{b^{3}}-\frac {2 \sqrt {e x +d}\, B a \,d^{3}}{b^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} A \,a^{2} e^{2}}{3 b^{3}}-\frac {4 \left (e x +d \right )^{\frac {3}{2}} A a d e}{3 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} A \,d^{2}}{3 b}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,a^{3} e^{2}}{3 b^{4}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} d e}{3 b^{3}}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B a \,d^{2}}{3 b^{2}}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} A a e}{5 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} A d}{5 b}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} B \,a^{2} e}{5 b^{3}}-\frac {2 \left (e x +d \right )^{\frac {5}{2}} B a d}{5 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {7}{2}} A}{7 b}-\frac {2 \left (e x +d \right )^{\frac {7}{2}} B a}{7 b^{2}}+\frac {2 \left (e x +d \right )^{\frac {9}{2}} B}{9 b e} \end {gather*}
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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mupad [B] time = 0.12, size = 426, normalized size = 2.15 \begin {gather*} \left (\frac {2\,A\,e-2\,B\,d}{7\,b\,e}-\frac {2\,B\,\left (a\,e^2-b\,d\,e\right )}{7\,b^2\,e^2}\right )\,{\left (d+e\,x\right )}^{7/2}+\frac {2\,B\,{\left (d+e\,x\right )}^{9/2}}{9\,b\,e}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}\,\sqrt {d+e\,x}}{-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}\right )\,\left (A\,b-B\,a\right )\,{\left (a\,e-b\,d\right )}^{7/2}}{b^{11/2}}+\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 )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,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 )}^3\,\sqrt {d+e\,x}}{b^3\,e^3}-\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 )}^{5/2}}{5\,b\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 109.54, size = 337, normalized size = 1.70 \begin {gather*} \frac {2 B \left (d + e x\right )^{\frac {9}{2}}}{9 b e} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (2 A b - 2 B a\right )}{7 b^{2}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{5 b^{3}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \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 )}{3 b^{4}} + \frac {\sqrt {d + e x} \left (- 2 A a^{3} b e^{3} + 6 A a^{2} b^{2} d e^{2} - 6 A a b^{3} d^{2} e + 2 A b^{4} d^{3} + 2 B a^{4} e^{3} - 6 B a^{3} b d e^{2} + 6 B a^{2} b^{2} d^{2} e - 2 B a b^{3} d^{3}\right )}{b^{5}} - \frac {2 \left (- A b + B a\right ) \left (a e - b d\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{6} \sqrt {\frac {a e - b d}{b}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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