Optimal. Leaf size=313 \[ -\frac {7 e^4 (-9 a B e-A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}-\frac {7 e^3 \sqrt {d+e x} (-9 a B e-A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac {7 e^2 (d+e x)^{3/2} (-9 a B e-A b e+10 b B d)}{192 b^4 (a+b x)^2 (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e-A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{7/2} (-9 a B e-A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
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Rubi [A] time = 0.26, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 78, 47, 63, 208} \begin {gather*} -\frac {7 e^2 (d+e x)^{3/2} (-9 a B e-A b e+10 b B d)}{192 b^4 (a+b x)^2 (b d-a e)}-\frac {7 e^3 \sqrt {d+e x} (-9 a B e-A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac {7 e^4 (-9 a B e-A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}-\frac {(d+e x)^{7/2} (-9 a B e-A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-9 a B e-A b e+10 b B d)}{240 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-A b e-9 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(7 e (10 b B d-A b e-9 a B e)) \int \frac {(d+e x)^{5/2}}{(a+b x)^4} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^2 (10 b B d-A b e-9 a B e)\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^3} \, dx}{96 b^3 (b d-a e)}\\ &=-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^3 (10 b B d-A b e-9 a B e)\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac {7 e^3 (10 b B d-A b e-9 a B e) \sqrt {d+e x}}{128 b^5 (b d-a e) (a+b x)}-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^4 (10 b B d-A b e-9 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^5 (b d-a e)}\\ &=-\frac {7 e^3 (10 b B d-A b e-9 a B e) \sqrt {d+e x}}{128 b^5 (b d-a e) (a+b x)}-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (7 e^3 (10 b B d-A b e-9 a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^5 (b d-a e)}\\ &=-\frac {7 e^3 (10 b B d-A b e-9 a B e) \sqrt {d+e x}}{128 b^5 (b d-a e) (a+b x)}-\frac {7 e^2 (10 b B d-A b e-9 a B e) (d+e x)^{3/2}}{192 b^4 (b d-a e) (a+b x)^2}-\frac {7 e (10 b B d-A b e-9 a B e) (d+e x)^{5/2}}{240 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-A b e-9 a B e) (d+e x)^{7/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{9/2}}{5 b (b d-a e) (a+b x)^5}-\frac {7 e^4 (10 b B d-A b e-9 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} (b d-a e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 260, normalized size = 0.83 \begin {gather*} \frac {\frac {(a+b x) (9 a B e+A b e-10 b B d) \left (48 b^4 (d+e x)^4 \sqrt {a e-b d}+56 b^3 e (a+b x) (d+e x)^3 \sqrt {a e-b d}+70 b^2 e^2 (a+b x)^2 (d+e x)^2 \sqrt {a e-b d}-105 \sqrt {b} e^4 (a+b x)^4 \sqrt {d+e x} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )+105 b e^3 (a+b x)^3 (d+e x) \sqrt {a e-b d}\right )}{\sqrt {a e-b d}}-384 b^5 (d+e x)^5 (A b-a B)}{1920 b^6 (a+b x)^5 \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 4.93, size = 676, normalized size = 2.16 \begin {gather*} -\frac {e^4 \sqrt {d+e x} \left (945 a^5 B e^5+105 a^4 A b e^5+4410 a^4 b B e^4 (d+e x)-4830 a^4 b B d e^4+490 a^3 A b^2 e^4 (d+e x)-420 a^3 A b^2 d e^4+9870 a^3 b^2 B d^2 e^3+8064 a^3 b^2 B e^3 (d+e x)^2-18130 a^3 b^2 B d e^3 (d+e x)+630 a^2 A b^3 d^2 e^3+896 a^2 A b^3 e^3 (d+e x)^2-1470 a^2 A b^3 d e^3 (d+e x)-10080 a^2 b^3 B d^3 e^2+27930 a^2 b^3 B d^2 e^2 (d+e x)+7110 a^2 b^3 B e^2 (d+e x)^3-25088 a^2 b^3 B d e^2 (d+e x)^2-420 a A b^4 d^3 e^2+1470 a A b^4 d^2 e^2 (d+e x)+790 a A b^4 e^2 (d+e x)^3-1792 a A b^4 d e^2 (d+e x)^2+5145 a b^4 B d^4 e-19110 a b^4 B d^3 e (d+e x)+25984 a b^4 B d^2 e (d+e x)^2+2895 a b^4 B e (d+e x)^4-15010 a b^4 B d e (d+e x)^3+105 A b^5 d^4 e-490 A b^5 d^3 e (d+e x)+896 A b^5 d^2 e (d+e x)^2-105 A b^5 e (d+e x)^4-790 A b^5 d e (d+e x)^3-1050 b^5 B d^5+4900 b^5 B d^4 (d+e x)-8960 b^5 B d^3 (d+e x)^2+7900 b^5 B d^2 (d+e x)^3-2790 b^5 B d (d+e x)^4\right )}{1920 b^5 (b d-a e) (-a e-b (d+e x)+b d)^5}-\frac {7 \left (-9 a B e^5-A b e^5+10 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{11/2} (b d-a e) \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 2041, normalized size = 6.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 781, normalized size = 2.50 \begin {gather*} \frac {7 \, {\left (10 \, B b d e^{4} - 9 \, B a e^{5} - A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d - a b^{5} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {2790 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 7900 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 8960 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 4900 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 1050 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 2895 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} + 105 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 15010 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 790 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 25984 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 19110 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 490 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 5145 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} - 105 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 7110 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 790 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 25088 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 27930 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 10080 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} + 420 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} - 8064 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 18130 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 9870 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} - 630 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} - 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 490 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} + 4830 \, \sqrt {x e + d} B a^{4} b d e^{8} + 420 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} - 945 \, \sqrt {x e + d} B a^{5} e^{9} - 105 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d - a b^{5} e\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 959, normalized size = 3.06
result too large to display
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 = 0.44, size = 594, normalized size = 1.90 \begin {gather*} \frac {7\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (A\,b\,e+9\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (A\,b\,e+9\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{11/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {79\,{\left (d+e\,x\right )}^{7/2}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^2}+\frac {7\,\sqrt {d+e\,x}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{128\,b^5}+\frac {7\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,b^3}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5-193\,B\,a\,e^5+186\,B\,b\,d\,e^4\right )}{128\,b\,\left (a\,e-b\,d\right )}+\frac {49\,{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )\,\left (A\,b\,e^5+9\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^4}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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