Optimal. Leaf size=256 \[ -\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
[Out]
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Rubi [A] time = 0.564854, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}}+\frac{\sqrt{d+e x} (b d-a e)^2 (-9 a B e+7 A b e+2 b B d)}{b^5}+\frac{(d+e x)^{3/2} (b d-a e) (-9 a B e+7 A b e+2 b B d)}{3 b^4}+\frac{(d+e x)^{5/2} (-9 a B e+7 A b e+2 b B d)}{5 b^3}+\frac{(d+e x)^{7/2} (-9 a B e+7 A b e+2 b B d)}{7 b^2 (b d-a e)}-\frac{(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 62.3861, size = 245, normalized size = 0.96 \[ \frac{\left (d + e x\right )^{\frac{9}{2}} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{7}{2}} \left (7 A b e - 9 B a e + 2 B b d\right )}{7 b^{2} \left (a e - b d\right )} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (7 A b e - 9 B a e + 2 B b d\right )}{5 b^{3}} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (7 A b e - 9 B a e + 2 B b d\right )}{3 b^{4}} + \frac{\sqrt{d + e x} \left (a e - b d\right )^{2} \left (7 A b e - 9 B a e + 2 B b d\right )}{b^{5}} - \frac{\left (a e - b d\right )^{\frac{5}{2}} \left (7 A b e - 9 B a e + 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 1.31201, size = 252, normalized size = 0.98 \[ \frac{\sqrt{d+e x} \left (-840 a^3 B e^3+2 b e x \left (105 a^2 B e^2-14 a b e (5 A e+16 B d)+2 b^2 d (56 A e+61 B d)\right )+210 a^2 b e^2 (3 A e+10 B d)+6 b^2 e^2 x^2 (-14 a B e+7 A b e+22 b B d)-56 a b^2 d e (25 A e+29 B d)-\frac{105 (A b-a B) (b d-a e)^3}{a+b x}+4 b^3 d^2 (203 A e+88 B d)+30 b^3 B e^3 x^3\right )}{105 b^5}-\frac{(b d-a e)^{5/2} (-9 a B e+7 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^2,x]
[Out]
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Maple [B] time = 0.031, size = 915, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(7/2)/(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231991, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(7/2)/(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(7/2)/(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.231603, size = 815, normalized size = 3.18 \[ \frac{{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{5}} + \frac{\sqrt{x e + d} B a b^{3} d^{3} e - \sqrt{x e + d} A b^{4} d^{3} e - 3 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt{x e + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt{x e + d} B a^{3} b d e^{3} - 3 \, \sqrt{x e + d} A a^{2} b^{2} d e^{3} - \sqrt{x e + d} B a^{4} e^{4} + \sqrt{x e + d} A a^{3} b e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{5}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{12} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{12} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{12} d^{2} + 105 \, \sqrt{x e + d} B b^{12} d^{3} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{11} e + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{12} e - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{11} d e + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{12} d e - 630 \, \sqrt{x e + d} B a b^{11} d^{2} e + 315 \, \sqrt{x e + d} A b^{12} d^{2} e + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{10} e^{2} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{11} e^{2} + 945 \, \sqrt{x e + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt{x e + d} A a b^{11} d e^{2} - 420 \, \sqrt{x e + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt{x e + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(7/2)/(b*x + a)^2,x, algorithm="giac")
[Out]