Optimal. Leaf size=452 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \]
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Rubi [A] time = 0.21, antiderivative size = 452, 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)^{13/2} (-5 a B e-A b e+6 b B d)}{13 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{11 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{7 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (A+B x) \sqrt {d+e x} \, 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) \sqrt {d+e x}}{e^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{3/2}}{e^6}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{5/2}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{11/2}}{e^6}+\frac {b^{10} B (d+e x)^{13/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 (B d-A e) (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 d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 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)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 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)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 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)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (-3465 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+20475 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-50050 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+32175 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-9009 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+15015 (b d-a e)^5 (B d-A e)+3003 b^5 B (d+e x)^6\right )}{45045 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 53.72, size = 812, normalized size = 1.80 \begin {gather*} \frac {2 (d+e x)^{3/2} \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (15015 b^5 B d^6-15015 A b^5 e d^5-75075 a b^4 B e d^5-54054 b^5 B (d+e x) d^5+75075 a A b^4 e^2 d^4+150150 a^2 b^3 B e^2 d^4+96525 b^5 B (d+e x)^2 d^4+45045 A b^5 e (d+e x) d^4+225225 a b^4 B e (d+e x) d^4-150150 a^2 A b^3 e^3 d^3-150150 a^3 b^2 B e^3 d^3-100100 b^5 B (d+e x)^3 d^3-64350 A b^5 e (d+e x)^2 d^3-321750 a b^4 B e (d+e x)^2 d^3-180180 a A b^4 e^2 (d+e x) d^3-360360 a^2 b^3 B e^2 (d+e x) d^3+150150 a^3 A b^2 e^4 d^2+75075 a^4 b B e^4 d^2+61425 b^5 B (d+e x)^4 d^2+50050 A b^5 e (d+e x)^3 d^2+250250 a b^4 B e (d+e x)^3 d^2+193050 a A b^4 e^2 (d+e x)^2 d^2+386100 a^2 b^3 B e^2 (d+e x)^2 d^2+270270 a^2 A b^3 e^3 (d+e x) d^2+270270 a^3 b^2 B e^3 (d+e x) d^2-75075 a^4 A b e^5 d-15015 a^5 B e^5 d-20790 b^5 B (d+e x)^5 d-20475 A b^5 e (d+e x)^4 d-102375 a b^4 B e (d+e x)^4 d-100100 a A b^4 e^2 (d+e x)^3 d-200200 a^2 b^3 B e^2 (d+e x)^3 d-193050 a^2 A b^3 e^3 (d+e x)^2 d-193050 a^3 b^2 B e^3 (d+e x)^2 d-180180 a^3 A b^2 e^4 (d+e x) d-90090 a^4 b B e^4 (d+e x) d+15015 a^5 A e^6+3003 b^5 B (d+e x)^6+3465 A b^5 e (d+e x)^5+17325 a b^4 B e (d+e x)^5+20475 a A b^4 e^2 (d+e x)^4+40950 a^2 b^3 B e^2 (d+e x)^4+50050 a^2 A b^3 e^3 (d+e x)^3+50050 a^3 b^2 B e^3 (d+e x)^3+64350 a^3 A b^2 e^4 (d+e x)^2+32175 a^4 b B e^4 (d+e x)^2+45045 a^4 A b e^5 (d+e x)+9009 a^5 B e^5 (d+e x)\right )}{45045 e^6 (a e+b x e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 702, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (3003 \, B b^{5} e^{7} x^{7} + 1024 \, B b^{5} d^{7} + 15015 \, A a^{5} d e^{6} - 1280 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6} e + 8320 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} e^{2} - 22880 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e^{3} + 17160 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{4} - 6006 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{5} + 231 \, {\left (B b^{5} d e^{6} + 15 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{7}\right )} x^{6} - 63 \, {\left (4 \, B b^{5} d^{2} e^{5} - 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{6} - 325 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{7}\right )} x^{5} + 35 \, {\left (8 \, B b^{5} d^{3} e^{4} - 10 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{5} + 65 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{6} + 1430 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{7}\right )} x^{4} - 5 \, {\left (64 \, B b^{5} d^{4} e^{3} - 80 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{4} + 520 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{5} - 1430 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{6} - 6435 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{7}\right )} x^{3} + 3 \, {\left (128 \, B b^{5} d^{5} e^{2} - 160 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{3} + 1040 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{4} - 2860 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{5} + 2145 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{6} + 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{7}\right )} x^{2} - {\left (512 \, B b^{5} d^{6} e - 15015 \, A a^{5} e^{7} - 640 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e^{2} + 4160 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{3} - 11440 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{4} + 8580 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{5} - 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 1682, normalized size = 3.72
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 = 1.52 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 B \,b^{5} e^{6} x^{6}+3465 A \,b^{5} e^{6} x^{5}+17325 B a \,b^{4} e^{6} x^{5}-2772 B \,b^{5} d \,e^{5} x^{5}+20475 A a \,b^{4} e^{6} x^{4}-3150 A \,b^{5} d \,e^{5} x^{4}+40950 B \,a^{2} b^{3} e^{6} x^{4}-15750 B a \,b^{4} d \,e^{5} x^{4}+2520 B \,b^{5} d^{2} e^{4} x^{4}+50050 A \,a^{2} b^{3} e^{6} x^{3}-18200 A a \,b^{4} d \,e^{5} x^{3}+2800 A \,b^{5} d^{2} e^{4} x^{3}+50050 B \,a^{3} b^{2} e^{6} x^{3}-36400 B \,a^{2} b^{3} d \,e^{5} x^{3}+14000 B a \,b^{4} d^{2} e^{4} x^{3}-2240 B \,b^{5} d^{3} e^{3} x^{3}+64350 A \,a^{3} b^{2} e^{6} x^{2}-42900 A \,a^{2} b^{3} d \,e^{5} x^{2}+15600 A a \,b^{4} d^{2} e^{4} x^{2}-2400 A \,b^{5} d^{3} e^{3} x^{2}+32175 B \,a^{4} b \,e^{6} x^{2}-42900 B \,a^{3} b^{2} d \,e^{5} x^{2}+31200 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-12000 B a \,b^{4} d^{3} e^{3} x^{2}+1920 B \,b^{5} d^{4} e^{2} x^{2}+45045 A \,a^{4} b \,e^{6} x -51480 A \,a^{3} b^{2} d \,e^{5} x +34320 A \,a^{2} b^{3} d^{2} e^{4} x -12480 A a \,b^{4} d^{3} e^{3} x +1920 A \,b^{5} d^{4} e^{2} x +9009 B \,a^{5} e^{6} x -25740 B \,a^{4} b d \,e^{5} x +34320 B \,a^{3} b^{2} d^{2} e^{4} x -24960 B \,a^{2} b^{3} d^{3} e^{3} x +9600 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +15015 A \,a^{5} e^{6}-30030 A \,a^{4} b d \,e^{5}+34320 A \,a^{3} b^{2} d^{2} e^{4}-22880 A \,a^{2} b^{3} d^{3} e^{3}+8320 A a \,b^{4} d^{4} e^{2}-1280 A \,b^{5} d^{5} e -6006 B \,a^{5} d \,e^{5}+17160 B \,a^{4} b \,d^{2} e^{4}-22880 B \,a^{3} b^{2} d^{3} e^{3}+16640 B \,a^{2} b^{3} d^{4} e^{2}-6400 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}}}{45045 \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 [B] time = 0.81, size = 760, normalized size = 1.68 \begin {gather*} \frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d} A}{9009 \, e^{6}} + \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} + 1024 \, b^{5} d^{7} - 6400 \, a b^{4} d^{6} e + 16640 \, a^{2} b^{3} d^{5} e^{2} - 22880 \, a^{3} b^{2} d^{4} e^{3} + 17160 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} + 231 \, {\left (b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{5} d^{2} e^{5} - 25 \, a b^{4} d e^{6} - 650 \, a^{2} b^{3} e^{7}\right )} x^{5} + 70 \, {\left (4 \, b^{5} d^{3} e^{4} - 25 \, a b^{4} d^{2} e^{5} + 65 \, a^{2} b^{3} d e^{6} + 715 \, a^{3} b^{2} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{5} d^{4} e^{3} - 400 \, a b^{4} d^{3} e^{4} + 1040 \, a^{2} b^{3} d^{2} e^{5} - 1430 \, a^{3} b^{2} d e^{6} - 6435 \, a^{4} b e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{5} e^{2} - 800 \, a b^{4} d^{4} e^{3} + 2080 \, a^{2} b^{3} d^{3} e^{4} - 2860 \, a^{3} b^{2} d^{2} e^{5} + 2145 \, a^{4} b d e^{6} + 3003 \, a^{5} e^{7}\right )} x^{2} - {\left (512 \, b^{5} d^{6} e - 3200 \, a b^{4} d^{5} e^{2} + 8320 \, a^{2} b^{3} d^{4} e^{3} - 11440 \, a^{3} b^{2} d^{3} e^{4} + 8580 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d} B}{45045 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B x\right ) \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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