Optimal. Leaf size=42 \[ \frac{2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (a+b x)^{9/2}}{9 b^2} \]
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Rubi [A] time = 0.044717, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (a+b x)^{9/2}}{9 b^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(5/2)*(A + B*x),x]
[Out]
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Rubi in Sympy [A] time = 7.92321, size = 37, normalized size = 0.88 \[ \frac{2 B \left (a + b x\right )^{\frac{9}{2}}}{9 b^{2}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(B*x+A),x)
[Out]
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Mathematica [A] time = 0.0489212, size = 30, normalized size = 0.71 \[ \frac{2 (a+b x)^{7/2} (-2 a B+9 A b+7 b B x)}{63 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(5/2)*(A + B*x),x]
[Out]
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Maple [A] time = 0.006, size = 27, normalized size = 0.6 \[{\frac{14\,bBx+18\,Ab-4\,Ba}{63\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(B*x+A),x)
[Out]
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Maxima [A] time = 1.34411, size = 45, normalized size = 1.07 \[ \frac{2 \,{\left (7 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 9 \,{\left (B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{63 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207943, size = 126, normalized size = 3. \[ \frac{2 \,{\left (7 \, B b^{4} x^{4} - 2 \, B a^{4} + 9 \, A a^{3} b +{\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{2} +{\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{63 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.38216, size = 194, normalized size = 4.62 \[ \begin{cases} \frac{2 A a^{3} \sqrt{a + b x}}{7 b} + \frac{6 A a^{2} x \sqrt{a + b x}}{7} + \frac{6 A a b x^{2} \sqrt{a + b x}}{7} + \frac{2 A b^{2} x^{3} \sqrt{a + b x}}{7} - \frac{4 B a^{4} \sqrt{a + b x}}{63 b^{2}} + \frac{2 B a^{3} x \sqrt{a + b x}}{63 b} + \frac{10 B a^{2} x^{2} \sqrt{a + b x}}{21} + \frac{38 B a b x^{3} \sqrt{a + b x}}{63} + \frac{2 B b^{2} x^{4} \sqrt{a + b x}}{9} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (A x + \frac{B x^{2}}{2}\right ) & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(5/2)*(B*x+A),x)
[Out]
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GIAC/XCAS [A] time = 0.212739, size = 308, normalized size = 7.33 \[ \frac{2 \,{\left (105 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} + 42 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A a + \frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} B a^{2}}{b} + \frac{6 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} B a}{b^{13}} + \frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} A}{b^{12}} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} B}{b^{25}}\right )}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2),x, algorithm="giac")
[Out]