Optimal. Leaf size=446 \[ -\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)} \]
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Rubi [A] time = 0.22, 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)^{9/2} (-5 a B e-A b e+6 b B d)}{9 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x)}-\frac {4 b^2 \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+2 b B d)}{e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4 (-a B e-5 A b e+6 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)^5 (B d-A e)}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 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)^{3/2}} \, 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)^{3/2}} \, 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)^{3/2}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 \sqrt {d+e x}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) \sqrt {d+e x}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{3/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{5/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{7/2}}{e^6}+\frac {b^{10} B (d+e x)^{9/2}}{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}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {2 (b d-a e)^4 (6 b B d-5 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 {10 b (b d-a e)^3 (3 b B d-2 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 {4 b^2 (b d-a e)^2 (2 b B d-A b e-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 {10 b^3 (b d-a e) (3 b B d-A b e-2 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^4 (6 b B d-A b e-5 a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 239, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-77 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+495 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-1386 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+1155 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-693 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)-693 (b d-a e)^5 (B d-A e)+63 b^5 B (d+e x)^6\right )}{693 e^7 (a+b x) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 38.16, size = 812, normalized size = 1.82 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (-693 b^5 B d^6+693 A b^5 e d^5+3465 a b^4 B e d^5-4158 b^5 B (d+e x) d^5-3465 a A b^4 e^2 d^4-6930 a^2 b^3 B e^2 d^4+3465 b^5 B (d+e x)^2 d^4+3465 A b^5 e (d+e x) d^4+17325 a b^4 B e (d+e x) d^4+6930 a^2 A b^3 e^3 d^3+6930 a^3 b^2 B e^3 d^3-2772 b^5 B (d+e x)^3 d^3-2310 A b^5 e (d+e x)^2 d^3-11550 a b^4 B e (d+e x)^2 d^3-13860 a A b^4 e^2 (d+e x) d^3-27720 a^2 b^3 B e^2 (d+e x) d^3-6930 a^3 A b^2 e^4 d^2-3465 a^4 b B e^4 d^2+1485 b^5 B (d+e x)^4 d^2+1386 A b^5 e (d+e x)^3 d^2+6930 a b^4 B e (d+e x)^3 d^2+6930 a A b^4 e^2 (d+e x)^2 d^2+13860 a^2 b^3 B e^2 (d+e x)^2 d^2+20790 a^2 A b^3 e^3 (d+e x) d^2+20790 a^3 b^2 B e^3 (d+e x) d^2+3465 a^4 A b e^5 d+693 a^5 B e^5 d-462 b^5 B (d+e x)^5 d-495 A b^5 e (d+e x)^4 d-2475 a b^4 B e (d+e x)^4 d-2772 a A b^4 e^2 (d+e x)^3 d-5544 a^2 b^3 B e^2 (d+e x)^3 d-6930 a^2 A b^3 e^3 (d+e x)^2 d-6930 a^3 b^2 B e^3 (d+e x)^2 d-13860 a^3 A b^2 e^4 (d+e x) d-6930 a^4 b B e^4 (d+e x) d-693 a^5 A e^6+63 b^5 B (d+e x)^6+77 A b^5 e (d+e x)^5+385 a b^4 B e (d+e x)^5+495 a A b^4 e^2 (d+e x)^4+990 a^2 b^3 B e^2 (d+e x)^4+1386 a^2 A b^3 e^3 (d+e x)^3+1386 a^3 b^2 B e^3 (d+e x)^3+2310 a^3 A b^2 e^4 (d+e x)^2+1155 a^4 b B e^4 (d+e x)^2+3465 a^4 A b e^5 (d+e x)+693 a^5 B e^5 (d+e x)\right )}{693 e^6 \sqrt {d+e x} (a e+b x e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 569, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (63 \, B b^{5} e^{6} x^{6} - 3072 \, B b^{5} d^{6} - 693 \, A a^{5} e^{6} + 2816 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 12672 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 22176 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 9240 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 7 \, {\left (12 \, B b^{5} d e^{5} - 11 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (24 \, B b^{5} d^{2} e^{4} - 22 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 99 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 2 \, {\left (96 \, B b^{5} d^{3} e^{3} - 88 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 396 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 693 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + {\left (384 \, B b^{5} d^{4} e^{2} - 352 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 1584 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 2772 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 1155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - {\left (1536 \, B b^{5} d^{5} e - 1408 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 6336 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 11088 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 4620 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 693 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{693 \, {\left (e^{8} x + d e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 1125, normalized size = 2.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 689, normalized size = 1.54 \begin {gather*} -\frac {2 \left (-63 B \,b^{5} e^{6} x^{6}-77 A \,b^{5} e^{6} x^{5}-385 B a \,b^{4} e^{6} x^{5}+84 B \,b^{5} d \,e^{5} x^{5}-495 A a \,b^{4} e^{6} x^{4}+110 A \,b^{5} d \,e^{5} x^{4}-990 B \,a^{2} b^{3} e^{6} x^{4}+550 B a \,b^{4} d \,e^{5} x^{4}-120 B \,b^{5} d^{2} e^{4} x^{4}-1386 A \,a^{2} b^{3} e^{6} x^{3}+792 A a \,b^{4} d \,e^{5} x^{3}-176 A \,b^{5} d^{2} e^{4} x^{3}-1386 B \,a^{3} b^{2} e^{6} x^{3}+1584 B \,a^{2} b^{3} d \,e^{5} x^{3}-880 B a \,b^{4} d^{2} e^{4} x^{3}+192 B \,b^{5} d^{3} e^{3} x^{3}-2310 A \,a^{3} b^{2} e^{6} x^{2}+2772 A \,a^{2} b^{3} d \,e^{5} x^{2}-1584 A a \,b^{4} d^{2} e^{4} x^{2}+352 A \,b^{5} d^{3} e^{3} x^{2}-1155 B \,a^{4} b \,e^{6} x^{2}+2772 B \,a^{3} b^{2} d \,e^{5} x^{2}-3168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1760 B a \,b^{4} d^{3} e^{3} x^{2}-384 B \,b^{5} d^{4} e^{2} x^{2}-3465 A \,a^{4} b \,e^{6} x +9240 A \,a^{3} b^{2} d \,e^{5} x -11088 A \,a^{2} b^{3} d^{2} e^{4} x +6336 A a \,b^{4} d^{3} e^{3} x -1408 A \,b^{5} d^{4} e^{2} x -693 B \,a^{5} e^{6} x +4620 B \,a^{4} b d \,e^{5} x -11088 B \,a^{3} b^{2} d^{2} e^{4} x +12672 B \,a^{2} b^{3} d^{3} e^{3} x -7040 B a \,b^{4} d^{4} e^{2} x +1536 B \,b^{5} d^{5} e x +693 A \,a^{5} e^{6}-6930 A \,a^{4} b d \,e^{5}+18480 A \,a^{3} b^{2} d^{2} e^{4}-22176 A \,a^{2} b^{3} d^{3} e^{3}+12672 A a \,b^{4} d^{4} e^{2}-2816 A \,b^{5} d^{5} e -1386 B \,a^{5} d \,e^{5}+9240 B \,a^{4} b \,d^{2} e^{4}-22176 B \,a^{3} b^{2} d^{3} e^{3}+25344 B \,a^{2} b^{3} d^{4} e^{2}-14080 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{693 \sqrt {e x +d}\, \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 = 0.66, size = 603, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} A}{63 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \, {\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} B}{693 \, \sqrt {e x + d} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 659, normalized size = 1.48 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {x^2\,\left (2310\,B\,a^4\,b\,e^6-5544\,B\,a^3\,b^2\,d\,e^5+4620\,A\,a^3\,b^2\,e^6+6336\,B\,a^2\,b^3\,d^2\,e^4-5544\,A\,a^2\,b^3\,d\,e^5-3520\,B\,a\,b^4\,d^3\,e^3+3168\,A\,a\,b^4\,d^2\,e^4+768\,B\,b^5\,d^4\,e^2-704\,A\,b^5\,d^3\,e^3\right )}{693\,b\,e^7}-\frac {-2772\,B\,a^5\,d\,e^5+1386\,A\,a^5\,e^6+18480\,B\,a^4\,b\,d^2\,e^4-13860\,A\,a^4\,b\,d\,e^5-44352\,B\,a^3\,b^2\,d^3\,e^3+36960\,A\,a^3\,b^2\,d^2\,e^4+50688\,B\,a^2\,b^3\,d^4\,e^2-44352\,A\,a^2\,b^3\,d^3\,e^3-28160\,B\,a\,b^4\,d^5\,e+25344\,A\,a\,b^4\,d^4\,e^2+6144\,B\,b^5\,d^6-5632\,A\,b^5\,d^5\,e}{693\,b\,e^7}+\frac {x^3\,\left (2772\,B\,a^3\,b^2\,e^6-3168\,B\,a^2\,b^3\,d\,e^5+2772\,A\,a^2\,b^3\,e^6+1760\,B\,a\,b^4\,d^2\,e^4-1584\,A\,a\,b^4\,d\,e^5-384\,B\,b^5\,d^3\,e^3+352\,A\,b^5\,d^2\,e^4\right )}{693\,b\,e^7}+\frac {2\,b^3\,x^5\,\left (11\,A\,b\,e+55\,B\,a\,e-12\,B\,b\,d\right )}{99\,e^2}+\frac {x\,\left (1386\,B\,a^5\,e^6-9240\,B\,a^4\,b\,d\,e^5+6930\,A\,a^4\,b\,e^6+22176\,B\,a^3\,b^2\,d^2\,e^4-18480\,A\,a^3\,b^2\,d\,e^5-25344\,B\,a^2\,b^3\,d^3\,e^3+22176\,A\,a^2\,b^3\,d^2\,e^4+14080\,B\,a\,b^4\,d^4\,e^2-12672\,A\,a\,b^4\,d^3\,e^3-3072\,B\,b^5\,d^5\,e+2816\,A\,b^5\,d^4\,e^2\right )}{693\,b\,e^7}+\frac {10\,b^2\,x^4\,\left (198\,B\,a^2\,e^2-110\,B\,a\,b\,d\,e+99\,A\,a\,b\,e^2+24\,B\,b^2\,d^2-22\,A\,b^2\,d\,e\right )}{693\,e^3}+\frac {2\,B\,b^4\,x^6}{11\,e}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \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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