Optimal. Leaf size=308 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)} \]
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Rubi [A] time = 0.14, antiderivative size = 308, 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^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (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)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (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) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^{3/2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^{3/2}}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{5/2}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{9/2}}{e^4}+\frac {b^6 B (d+e x)^{11/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {2 b (b d-a e) (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}}{3 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 163, normalized size = 0.53 \begin {gather*} \frac {2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{5/2} \left (-1365 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+5005 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-2145 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+3003 (b d-a e)^3 (B d-A e)+1155 b^3 B (d+e x)^4\right )}{15015 e^5 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 53.02, size = 374, normalized size = 1.21 \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (3003 a^3 A e^4+2145 a^3 B e^3 (d+e x)-3003 a^3 B d e^3+6435 a^2 A b e^3 (d+e x)-9009 a^2 A b d e^3+9009 a^2 b B d^2 e^2-12870 a^2 b B d e^2 (d+e x)+5005 a^2 b B e^2 (d+e x)^2+9009 a A b^2 d^2 e^2-12870 a A b^2 d e^2 (d+e x)+5005 a A b^2 e^2 (d+e x)^2-9009 a b^2 B d^3 e+19305 a b^2 B d^2 e (d+e x)-15015 a b^2 B d e (d+e x)^2+4095 a b^2 B e (d+e x)^3-3003 A b^3 d^3 e+6435 A b^3 d^2 e (d+e x)-5005 A b^3 d e (d+e x)^2+1365 A b^3 e (d+e x)^3+3003 b^3 B d^4-8580 b^3 B d^3 (d+e x)+10010 b^3 B d^2 (d+e x)^2-5460 b^3 B d (d+e x)^3+1155 b^3 B (d+e x)^4\right )}{15015 e^4 (a e+b e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 446, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left (1155 \, B b^{3} e^{6} x^{6} + 128 \, B b^{3} d^{6} + 3003 \, A a^{3} d^{2} e^{4} - 208 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e + 1144 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{2} - 858 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{3} + 105 \, {\left (14 \, B b^{3} d e^{5} + 13 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{6}\right )} x^{5} + 35 \, {\left (B b^{3} d^{2} e^{4} + 52 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{5} + 143 \, {\left (B a^{2} b + A a b^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{3} e^{3} - 13 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{4} - 1430 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{5} - 429 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{4} e^{2} + 1001 \, A a^{3} e^{6} - 26 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{3} + 143 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{4} + 1144 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{5} e - 6006 \, A a^{3} d e^{5} - 104 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{2} + 572 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{3} - 429 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 1524, normalized size = 4.95
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 = 317, normalized size = 1.03 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 b^{3} B \,x^{4} e^{4}+1365 A \,b^{3} e^{4} x^{3}+4095 B a \,b^{2} e^{4} x^{3}-840 B \,b^{3} d \,e^{3} x^{3}+5005 A a \,b^{2} e^{4} x^{2}-910 A \,b^{3} d \,e^{3} x^{2}+5005 B \,a^{2} b \,e^{4} x^{2}-2730 B a \,b^{2} d \,e^{3} x^{2}+560 B \,b^{3} d^{2} e^{2} x^{2}+6435 A \,a^{2} b \,e^{4} x -2860 A a \,b^{2} d \,e^{3} x +520 A \,b^{3} d^{2} e^{2} x +2145 B \,a^{3} e^{4} x -2860 B \,a^{2} b d \,e^{3} x +1560 B a \,b^{2} d^{2} e^{2} x -320 B \,b^{3} d^{3} e x +3003 A \,a^{3} e^{4}-2574 A \,a^{2} b d \,e^{3}+1144 A a \,b^{2} d^{2} e^{2}-208 A \,b^{3} d^{3} e -858 B \,a^{3} d \,e^{3}+1144 B \,a^{2} b \,d^{2} e^{2}-624 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15015 \left (b x +a \right )^{3} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 488, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} A}{1155 \, e^{4}} + \frac {2 \, {\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \, {\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \, {\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} B}{15015 \, e^{5}} \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 )\,{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \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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