Optimal. Leaf size=83 \[ -\frac {2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac {2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac {2 b B (d+e x)^{11/2}}{11 e^3} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {2 (d+e x)^{9/2} (-a B e-A b e+2 b B d)}{9 e^3}+\frac {2 (d+e x)^{7/2} (b d-a e) (B d-A e)}{7 e^3}+\frac {2 b B (d+e x)^{11/2}}{11 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x) (A+B x) (d+e x)^{5/2} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^{5/2}}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^2}+\frac {b B (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac {2 (b d-a e) (B d-A e) (d+e x)^{7/2}}{7 e^3}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{9/2}}{9 e^3}+\frac {2 b B (d+e x)^{11/2}}{11 e^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 70, normalized size = 0.84 \[ \frac {2 (d+e x)^{7/2} \left (11 a e (9 A e-2 B d+7 B e x)+11 A b e (7 e x-2 d)+b B \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 189, normalized size = 2.28 \[ \frac {2 \, {\left (63 \, B b e^{5} x^{5} + 8 \, B b d^{5} + 99 \, A a d^{3} e^{2} - 22 \, {\left (B a + A b\right )} d^{4} e + 7 \, {\left (23 \, B b d e^{4} + 11 \, {\left (B a + A b\right )} e^{5}\right )} x^{4} + {\left (113 \, B b d^{2} e^{3} + 99 \, A a e^{5} + 209 \, {\left (B a + A b\right )} d e^{4}\right )} x^{3} + 3 \, {\left (B b d^{3} e^{2} + 99 \, A a d e^{4} + 55 \, {\left (B a + A b\right )} d^{2} e^{3}\right )} x^{2} - {\left (4 \, B b d^{4} e - 297 \, A a d^{2} e^{3} - 11 \, {\left (B a + A b\right )} d^{3} e^{2}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.35, size = 778, normalized size = 9.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 73, normalized size = 0.88 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 B b \,x^{2} e^{2}+77 A b \,e^{2} x +77 B a \,e^{2} x -28 B b d e x +99 A a \,e^{2}-22 A b d e -22 B a d e +8 B b \,d^{2}\right )}{693 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 75, normalized size = 0.90 \[ \frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} B b - 77 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 99 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 80, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (63\,B\,b\,{\left (d+e\,x\right )}^2+99\,A\,a\,e^2+99\,B\,b\,d^2+77\,A\,b\,e\,\left (d+e\,x\right )+77\,B\,a\,e\,\left (d+e\,x\right )-154\,B\,b\,d\,\left (d+e\,x\right )-99\,A\,b\,d\,e-99\,B\,a\,d\,e\right )}{693\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.66, size = 476, normalized size = 5.73 \[ \begin {cases} \frac {2 A a d^{3} \sqrt {d + e x}}{7 e} + \frac {6 A a d^{2} x \sqrt {d + e x}}{7} + \frac {6 A a d e x^{2} \sqrt {d + e x}}{7} + \frac {2 A a e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {4 A b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 A b d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 A b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 A b d e x^{3} \sqrt {d + e x}}{63} + \frac {2 A b e^{2} x^{4} \sqrt {d + e x}}{9} - \frac {4 B a d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 B a d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 B a d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 B a d e x^{3} \sqrt {d + e x}}{63} + \frac {2 B a e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 B b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 B b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 B b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 B b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 B b d e x^{4} \sqrt {d + e x}}{99} + \frac {2 B b e^{2} x^{5} \sqrt {d + e x}}{11} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a x + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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