Optimal. Leaf size=446 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{3 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)} \]
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Rubi [A] time = 0.20, antiderivative size = 446, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{3 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{5/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{5/2}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{3/2}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 \sqrt {d+e x}}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) \sqrt {d+e x}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{3/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{5/2}}{e^6}+\frac {b^{10} B (d+e x)^{7/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {2 (b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{3/2}}+\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {2 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 239, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-9 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+63 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-210 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+315 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)+63 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-21 (b d-a e)^5 (B d-A e)+7 b^5 B (d+e x)^6\right )}{63 e^7 (a+b x) (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 30.08, size = 812, normalized size = 1.82 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (-21 b^5 B d^6+21 A b^5 e d^5+105 a b^4 B e d^5+378 b^5 B (d+e x) d^5-105 a A b^4 e^2 d^4-210 a^2 b^3 B e^2 d^4+945 b^5 B (d+e x)^2 d^4-315 A b^5 e (d+e x) d^4-1575 a b^4 B e (d+e x) d^4+210 a^2 A b^3 e^3 d^3+210 a^3 b^2 B e^3 d^3-420 b^5 B (d+e x)^3 d^3-630 A b^5 e (d+e x)^2 d^3-3150 a b^4 B e (d+e x)^2 d^3+1260 a A b^4 e^2 (d+e x) d^3+2520 a^2 b^3 B e^2 (d+e x) d^3-210 a^3 A b^2 e^4 d^2-105 a^4 b B e^4 d^2+189 b^5 B (d+e x)^4 d^2+210 A b^5 e (d+e x)^3 d^2+1050 a b^4 B e (d+e x)^3 d^2+1890 a A b^4 e^2 (d+e x)^2 d^2+3780 a^2 b^3 B e^2 (d+e x)^2 d^2-1890 a^2 A b^3 e^3 (d+e x) d^2-1890 a^3 b^2 B e^3 (d+e x) d^2+105 a^4 A b e^5 d+21 a^5 B e^5 d-54 b^5 B (d+e x)^5 d-63 A b^5 e (d+e x)^4 d-315 a b^4 B e (d+e x)^4 d-420 a A b^4 e^2 (d+e x)^3 d-840 a^2 b^3 B e^2 (d+e x)^3 d-1890 a^2 A b^3 e^3 (d+e x)^2 d-1890 a^3 b^2 B e^3 (d+e x)^2 d+1260 a^3 A b^2 e^4 (d+e x) d+630 a^4 b B e^4 (d+e x) d-21 a^5 A e^6+7 b^5 B (d+e x)^6+9 A b^5 e (d+e x)^5+45 a b^4 B e (d+e x)^5+63 a A b^4 e^2 (d+e x)^4+126 a^2 b^3 B e^2 (d+e x)^4+210 a^2 A b^3 e^3 (d+e x)^3+210 a^3 b^2 B e^3 (d+e x)^3+630 a^3 A b^2 e^4 (d+e x)^2+315 a^4 b B e^4 (d+e x)^2-315 a^4 A b e^5 (d+e x)-63 a^5 B e^5 (d+e x)\right )}{63 e^6 (d+e x)^{3/2} (a e+b x e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 581, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (7 \, B b^{5} e^{6} x^{6} + 1024 \, B b^{5} d^{6} - 21 \, A a^{5} e^{6} - 768 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 2688 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 3360 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 840 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 42 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \, {\left (4 \, B b^{5} d e^{5} - 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 3 \, {\left (8 \, B b^{5} d^{2} e^{4} - 6 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 21 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 2 \, {\left (32 \, B b^{5} d^{3} e^{3} - 24 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 84 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 3 \, {\left (128 \, B b^{5} d^{4} e^{2} - 96 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 336 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 420 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 105 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 3 \, {\left (512 \, B b^{5} d^{5} e - 384 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 1344 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 1680 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 420 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 21 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 1101, normalized size = 2.47
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 689, normalized size = 1.54 \begin {gather*} -\frac {2 \left (-7 B \,b^{5} e^{6} x^{6}-9 A \,b^{5} e^{6} x^{5}-45 B a \,b^{4} e^{6} x^{5}+12 B \,b^{5} d \,e^{5} x^{5}-63 A a \,b^{4} e^{6} x^{4}+18 A \,b^{5} d \,e^{5} x^{4}-126 B \,a^{2} b^{3} e^{6} x^{4}+90 B a \,b^{4} d \,e^{5} x^{4}-24 B \,b^{5} d^{2} e^{4} x^{4}-210 A \,a^{2} b^{3} e^{6} x^{3}+168 A a \,b^{4} d \,e^{5} x^{3}-48 A \,b^{5} d^{2} e^{4} x^{3}-210 B \,a^{3} b^{2} e^{6} x^{3}+336 B \,a^{2} b^{3} d \,e^{5} x^{3}-240 B a \,b^{4} d^{2} e^{4} x^{3}+64 B \,b^{5} d^{3} e^{3} x^{3}-630 A \,a^{3} b^{2} e^{6} x^{2}+1260 A \,a^{2} b^{3} d \,e^{5} x^{2}-1008 A a \,b^{4} d^{2} e^{4} x^{2}+288 A \,b^{5} d^{3} e^{3} x^{2}-315 B \,a^{4} b \,e^{6} x^{2}+1260 B \,a^{3} b^{2} d \,e^{5} x^{2}-2016 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1440 B a \,b^{4} d^{3} e^{3} x^{2}-384 B \,b^{5} d^{4} e^{2} x^{2}+315 A \,a^{4} b \,e^{6} x -2520 A \,a^{3} b^{2} d \,e^{5} x +5040 A \,a^{2} b^{3} d^{2} e^{4} x -4032 A a \,b^{4} d^{3} e^{3} x +1152 A \,b^{5} d^{4} e^{2} x +63 B \,a^{5} e^{6} x -1260 B \,a^{4} b d \,e^{5} x +5040 B \,a^{3} b^{2} d^{2} e^{4} x -8064 B \,a^{2} b^{3} d^{3} e^{3} x +5760 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +21 A \,a^{5} e^{6}+210 A \,a^{4} b d \,e^{5}-1680 A \,a^{3} b^{2} d^{2} e^{4}+3360 A \,a^{2} b^{3} d^{3} e^{3}-2688 A a \,b^{4} d^{4} e^{2}+768 A \,b^{5} d^{5} e +42 B \,a^{5} d \,e^{5}-840 B \,a^{4} b \,d^{2} e^{4}+3360 B \,a^{3} b^{2} d^{3} e^{3}-5376 B \,a^{2} b^{3} d^{4} e^{2}+3840 B a \,b^{4} d^{5} e -1024 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{5} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 625, normalized size = 1.40 \begin {gather*} \frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} A}{21 \, {\left (e^{7} x + d e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (7 \, b^{5} e^{6} x^{6} + 1024 \, b^{5} d^{6} - 3840 \, a b^{4} d^{5} e + 5376 \, a^{2} b^{3} d^{4} e^{2} - 3360 \, a^{3} b^{2} d^{3} e^{3} + 840 \, a^{4} b d^{2} e^{4} - 42 \, a^{5} d e^{5} - 3 \, {\left (4 \, b^{5} d e^{5} - 15 \, a b^{4} e^{6}\right )} x^{5} + 6 \, {\left (4 \, b^{5} d^{2} e^{4} - 15 \, a b^{4} d e^{5} + 21 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (32 \, b^{5} d^{3} e^{3} - 120 \, a b^{4} d^{2} e^{4} + 168 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{4} e^{2} - 480 \, a b^{4} d^{3} e^{3} + 672 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 105 \, a^{4} b e^{6}\right )} x^{2} + 3 \, {\left (512 \, b^{5} d^{5} e - 1920 \, a b^{4} d^{4} e^{2} + 2688 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 420 \, a^{4} b d e^{5} - 21 \, a^{5} e^{6}\right )} x\right )} B}{63 \, {\left (e^{8} x + d e^{7}\right )} \sqrt {e x + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.89, size = 695, normalized size = 1.56 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x^2\,\left (630\,B\,a^4\,b\,e^6-2520\,B\,a^3\,b^2\,d\,e^5+1260\,A\,a^3\,b^2\,e^6+4032\,B\,a^2\,b^3\,d^2\,e^4-2520\,A\,a^2\,b^3\,d\,e^5-2880\,B\,a\,b^4\,d^3\,e^3+2016\,A\,a\,b^4\,d^2\,e^4+768\,B\,b^5\,d^4\,e^2-576\,A\,b^5\,d^3\,e^3\right )}{63\,b\,e^8}-\frac {84\,B\,a^5\,d\,e^5+42\,A\,a^5\,e^6-1680\,B\,a^4\,b\,d^2\,e^4+420\,A\,a^4\,b\,d\,e^5+6720\,B\,a^3\,b^2\,d^3\,e^3-3360\,A\,a^3\,b^2\,d^2\,e^4-10752\,B\,a^2\,b^3\,d^4\,e^2+6720\,A\,a^2\,b^3\,d^3\,e^3+7680\,B\,a\,b^4\,d^5\,e-5376\,A\,a\,b^4\,d^4\,e^2-2048\,B\,b^5\,d^6+1536\,A\,b^5\,d^5\,e}{63\,b\,e^8}+\frac {x^3\,\left (420\,B\,a^3\,b^2\,e^6-672\,B\,a^2\,b^3\,d\,e^5+420\,A\,a^2\,b^3\,e^6+480\,B\,a\,b^4\,d^2\,e^4-336\,A\,a\,b^4\,d\,e^5-128\,B\,b^5\,d^3\,e^3+96\,A\,b^5\,d^2\,e^4\right )}{63\,b\,e^8}+\frac {2\,b^3\,x^5\,\left (3\,A\,b\,e+15\,B\,a\,e-4\,B\,b\,d\right )}{21\,e^3}-\frac {x\,\left (126\,B\,a^5\,e^6-2520\,B\,a^4\,b\,d\,e^5+630\,A\,a^4\,b\,e^6+10080\,B\,a^3\,b^2\,d^2\,e^4-5040\,A\,a^3\,b^2\,d\,e^5-16128\,B\,a^2\,b^3\,d^3\,e^3+10080\,A\,a^2\,b^3\,d^2\,e^4+11520\,B\,a\,b^4\,d^4\,e^2-8064\,A\,a\,b^4\,d^3\,e^3-3072\,B\,b^5\,d^5\,e+2304\,A\,b^5\,d^4\,e^2\right )}{63\,b\,e^8}+\frac {2\,b^2\,x^4\,\left (42\,B\,a^2\,e^2-30\,B\,a\,b\,d\,e+21\,A\,a\,b\,e^2+8\,B\,b^2\,d^2-6\,A\,b^2\,d\,e\right )}{21\,e^4}+\frac {2\,B\,b^4\,x^6}{9\,e^2}\right )}{x^2\,\sqrt {d+e\,x}+\frac {a\,d\,\sqrt {d+e\,x}}{b\,e}+\frac {x\,\left (63\,a\,e^8+63\,b\,d\,e^7\right )\,\sqrt {d+e\,x}}{63\,b\,e^8}} \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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