Optimal. Leaf size=126 \[ -\frac {2 (d+e x)^{11/2} (-A c e-b B e+3 B c d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{9 e^4}-\frac {2 d (d+e x)^{7/2} (B d-A e) (c d-b e)}{7 e^4}+\frac {2 B c (d+e x)^{13/2}}{13 e^4} \]
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Rubi [A] time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{11/2} (-A c e-b B e+3 B c d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{9 e^4}-\frac {2 d (d+e x)^{7/2} (B d-A e) (c d-b e)}{7 e^4}+\frac {2 B c (d+e x)^{13/2}}{13 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e) (d+e x)^{5/2}}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{7/2}}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^{9/2}}{e^3}+\frac {B c (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac {2 d (B d-A e) (c d-b e) (d+e x)^{7/2}}{7 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{9/2}}{9 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{11/2}}{11 e^4}+\frac {2 B c (d+e x)^{13/2}}{13 e^4}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 113, normalized size = 0.90 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (13 A e \left (11 b e (7 e x-2 d)+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+B \left (13 b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+c \left (-48 d^3+168 d^2 e x-378 d e^2 x^2+693 e^3 x^3\right )\right )\right )}{9009 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 141, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (1001 A b e^2 (d+e x)-1287 A b d e^2+1287 A c d^2 e-2002 A c d e (d+e x)+819 A c e (d+e x)^2+1287 b B d^2 e-2002 b B d e (d+e x)+819 b B e (d+e x)^2-1287 B c d^3+3003 B c d^2 (d+e x)-2457 B c d (d+e x)^2+693 B c (d+e x)^3\right )}{9009 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 230, normalized size = 1.83 \begin {gather*} \frac {2 \, {\left (693 \, B c e^{6} x^{6} - 48 \, B c d^{6} - 286 \, A b d^{4} e^{2} + 104 \, {\left (B b + A c\right )} d^{5} e + 63 \, {\left (27 \, B c d e^{5} + 13 \, {\left (B b + A c\right )} e^{6}\right )} x^{5} + 7 \, {\left (159 \, B c d^{2} e^{4} + 143 \, A b e^{6} + 299 \, {\left (B b + A c\right )} d e^{5}\right )} x^{4} + {\left (15 \, B c d^{3} e^{3} + 2717 \, A b d e^{5} + 1469 \, {\left (B b + A c\right )} d^{2} e^{4}\right )} x^{3} - 3 \, {\left (6 \, B c d^{4} e^{2} - 715 \, A b d^{2} e^{4} - 13 \, {\left (B b + A c\right )} d^{3} e^{3}\right )} x^{2} + {\left (24 \, B c d^{5} e + 143 \, A b d^{3} e^{3} - 52 \, {\left (B b + A c\right )} d^{4} e^{2}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 999, normalized size = 7.93
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 = 121, normalized size = 0.96 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (-693 B c \,x^{3} e^{3}-819 A c \,e^{3} x^{2}-819 B b \,e^{3} x^{2}+378 B c d \,e^{2} x^{2}-1001 A b \,e^{3} x +364 A c d \,e^{2} x +364 B b d \,e^{2} x -168 B c \,d^{2} e x +286 A b d \,e^{2}-104 A c \,d^{2} e -104 B b \,d^{2} e +48 B c \,d^{3}\right )}{9009 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 112, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (693 \, {\left (e x + d\right )}^{\frac {13}{2}} B c - 819 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 1287 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{9009 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 111, normalized size = 0.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{9\,e^4}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{11\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.64, size = 581, normalized size = 4.61 \begin {gather*} \begin {cases} - \frac {4 A b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 A b d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 A b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 A b d e x^{3} \sqrt {d + e x}}{63} + \frac {2 A b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 A c d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 A c d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 A c d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 A c d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 A c d e x^{4} \sqrt {d + e x}}{99} + \frac {2 A c e^{2} x^{5} \sqrt {d + e x}}{11} + \frac {16 B b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 B b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 B b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 B b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 B b d e x^{4} \sqrt {d + e x}}{99} + \frac {2 B b e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 B c d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 B c d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 B c d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 B c d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 B c d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 B c d e x^{5} \sqrt {d + e x}}{143} + \frac {2 B c e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\frac {A b x^{2}}{2} + \frac {A c x^{3}}{3} + \frac {B b x^{3}}{3} + \frac {B c x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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