Optimal. Leaf size=352 \[ -\frac {63 e^4 (-11 a B e+A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2} \sqrt {b d-a e}}+\frac {63 e^4 \sqrt {d+e x} (-11 a B e+A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac {21 e^3 (d+e x)^{3/2} (-11 a B e+A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+A b e+10 b B d)}{80 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 50, 63, 208} \begin {gather*} -\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac {21 e^3 (d+e x)^{3/2} (-11 a B e+A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}+\frac {63 e^4 \sqrt {d+e x} (-11 a B e+A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac {63 e^4 (-11 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^{13/2} \sqrt {b d-a e}}-\frac {(d+e x)^{9/2} (-11 a B e+A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+A b e+10 b B d)}{80 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 47
Rule 50
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{9/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d+A b e-11 a B e) \int \frac {(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(9 e (10 b B d+A b e-11 a B e)) \int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (21 e^2 (10 b B d+A b e-11 a B e)\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3 (b d-a e)}\\ &=-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (21 e^3 (10 b B d+A b e-11 a B e)\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (63 e^4 (10 b B d+A b e-11 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^5 (b d-a e)}\\ &=\frac {63 e^4 (10 b B d+A b e-11 a B e) \sqrt {d+e x}}{128 b^6 (b d-a e)}-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (63 e^4 (10 b B d+A b e-11 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^6}\\ &=\frac {63 e^4 (10 b B d+A b e-11 a B e) \sqrt {d+e x}}{128 b^6 (b d-a e)}-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (63 e^3 (10 b B d+A b e-11 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^6}\\ &=\frac {63 e^4 (10 b B d+A b e-11 a B e) \sqrt {d+e x}}{128 b^6 (b d-a e)}-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}-\frac {63 e^4 (10 b B d+A b e-11 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2} \sqrt {b d-a e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 99, normalized size = 0.28 \begin {gather*} \frac {(d+e x)^{11/2} \left (\frac {11 (a B-A b)}{(a+b x)^5}-\frac {e^4 (-11 a B e+A b e+10 b B d) \, _2F_1\left (5,\frac {11}{2};\frac {13}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}\right )}{55 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 5.33, size = 667, normalized size = 1.89 \begin {gather*} -\frac {e^4 \sqrt {d+e x} \left (-3465 a^5 B e^5+315 a^4 A b e^5-16170 a^4 b B e^4 (d+e x)+17010 a^4 b B d e^4+1470 a^3 A b^2 e^4 (d+e x)-1260 a^3 A b^2 d e^4-33390 a^3 b^2 B d^2 e^3-29568 a^3 b^2 B e^3 (d+e x)^2+63210 a^3 b^2 B d e^3 (d+e x)+1890 a^2 A b^3 d^2 e^3+2688 a^2 A b^3 e^3 (d+e x)^2-4410 a^2 A b^3 d e^3 (d+e x)+32760 a^2 b^3 B d^3 e^2-92610 a^2 b^3 B d^2 e^2 (d+e x)-26070 a^2 b^3 B e^2 (d+e x)^3+86016 a^2 b^3 B d e^2 (d+e x)^2-1260 a A b^4 d^3 e^2+4410 a A b^4 d^2 e^2 (d+e x)+2370 a A b^4 e^2 (d+e x)^3-5376 a A b^4 d e^2 (d+e x)^2-16065 a b^4 B d^4 e+60270 a b^4 B d^3 e (d+e x)-83328 a b^4 B d^2 e (d+e x)^2-10615 a b^4 B e (d+e x)^4+49770 a b^4 B d e (d+e x)^3+315 A b^5 d^4 e-1470 A b^5 d^3 e (d+e x)+2688 A b^5 d^2 e (d+e x)^2+965 A b^5 e (d+e x)^4-2370 A b^5 d e (d+e x)^3+3150 b^5 B d^5-14700 b^5 B d^4 (d+e x)+26880 b^5 B d^3 (d+e x)^2-23700 b^5 B d^2 (d+e x)^3-1280 b^5 B (d+e x)^5+9650 b^5 B d (d+e x)^4\right )}{640 b^6 (a e+b (d+e x)-b d)^5}-\frac {63 \left (-11 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^{13/2} \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 1955, normalized size = 5.55
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.30, size = 770, normalized size = 2.19 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{4}}{b^{6}} + \frac {63 \, {\left (10 \, B b d e^{4} - 11 \, 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 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {3250 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 10900 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 14080 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 8300 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 1870 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 4215 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} + 965 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 24170 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} - 2370 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 44928 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 34670 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} - 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 9665 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 315 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 13270 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} + 2370 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 47616 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} - 5376 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 54210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} + 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 19960 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 1260 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} - 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 37610 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} - 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 20590 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 1890 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} - 9770 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} + 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} + 10610 \, \sqrt {x e + d} B a^{4} b d e^{8} - 1260 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} - 2185 \, \sqrt {x e + d} B a^{5} e^{9} + 315 \, \sqrt {x e + d} A a^{4} b e^{9}}{640 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 1173, normalized size = 3.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.28, size = 838, normalized size = 2.38 \begin {gather*} \frac {2\,B\,e^4\,\sqrt {d+e\,x}}{b^6}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-\frac {131\,B\,a^3\,b^2\,e^7}{5}+\frac {372\,B\,a^2\,b^3\,d\,e^6}{5}+\frac {21\,A\,a^2\,b^3\,e^7}{5}-\frac {351\,B\,a\,b^4\,d^2\,e^5}{5}-\frac {42\,A\,a\,b^4\,d\,e^6}{5}+22\,B\,b^5\,d^3\,e^4+\frac {21\,A\,b^5\,d^2\,e^5}{5}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (\frac {977\,B\,a^4\,b\,e^8}{64}-\frac {3761\,B\,a^3\,b^2\,d\,e^7}{64}-\frac {147\,A\,a^3\,b^2\,e^8}{64}+\frac {5421\,B\,a^2\,b^3\,d^2\,e^6}{64}+\frac {441\,A\,a^2\,b^3\,d\,e^7}{64}-\frac {3467\,B\,a\,b^4\,d^3\,e^5}{64}-\frac {441\,A\,a\,b^4\,d^2\,e^6}{64}+\frac {415\,B\,b^5\,d^4\,e^4}{32}+\frac {147\,A\,b^5\,d^3\,e^5}{64}\right )+{\left (d+e\,x\right )}^{9/2}\,\left (\frac {193\,A\,b^5\,e^5}{128}+\frac {325\,B\,d\,b^5\,e^4}{64}-\frac {843\,B\,a\,b^4\,e^5}{128}\right )+\sqrt {d+e\,x}\,\left (-\frac {437\,B\,a^5\,e^9}{128}+\frac {1061\,B\,a^4\,b\,d\,e^8}{64}+\frac {63\,A\,a^4\,b\,e^9}{128}-\frac {2059\,B\,a^3\,b^2\,d^2\,e^7}{64}-\frac {63\,A\,a^3\,b^2\,d\,e^8}{32}+\frac {499\,B\,a^2\,b^3\,d^3\,e^6}{16}+\frac {189\,A\,a^2\,b^3\,d^2\,e^7}{64}-\frac {1933\,B\,a\,b^4\,d^4\,e^5}{128}-\frac {63\,A\,a\,b^4\,d^3\,e^6}{32}+\frac {187\,B\,b^5\,d^5\,e^4}{64}+\frac {63\,A\,b^5\,d^4\,e^5}{128}\right )-{\left (d+e\,x\right )}^{7/2}\,\left (\frac {1327\,B\,a^2\,b^3\,e^6}{64}-\frac {2417\,B\,a\,b^4\,d\,e^5}{64}-\frac {237\,A\,a\,b^4\,e^6}{64}+\frac {545\,B\,b^5\,d^2\,e^4}{32}+\frac {237\,A\,b^5\,d\,e^5}{64}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^7\,e^4-20\,a^3\,b^8\,d\,e^3+30\,a^2\,b^9\,d^2\,e^2-20\,a\,b^{10}\,d^3\,e+5\,b^{11}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^8\,e^3+30\,a^2\,b^9\,d\,e^2-30\,a\,b^{10}\,d^2\,e+10\,b^{11}\,d^3\right )+b^{11}\,{\left (d+e\,x\right )}^5-\left (5\,b^{11}\,d-5\,a\,b^{10}\,e\right )\,{\left (d+e\,x\right )}^4-b^{11}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^9\,e^2-20\,a\,b^{10}\,d\,e+10\,b^{11}\,d^2\right )+a^5\,b^6\,e^5-5\,a^4\,b^7\,d\,e^4-10\,a^2\,b^9\,d^3\,e^2+10\,a^3\,b^8\,d^2\,e^3+5\,a\,b^{10}\,d^4\,e}+\frac {63\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (A\,b\,e-11\,B\,a\,e+10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^5-11\,B\,a\,e^5+10\,B\,b\,d\,e^4\right )}\right )\,\left (A\,b\,e-11\,B\,a\,e+10\,B\,b\,d\right )}{128\,b^{13/2}\,\sqrt {a\,e-b\,d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________