Optimal. Leaf size=1128 \[ -\frac{2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 \sqrt{d g-c h} \sqrt{f g-e h} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{2 d \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B f g-B e h) c+2 B d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
[Out]
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Rubi [A] time = 9.95317, antiderivative size = 1119, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{2 (b B-2 a C) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} b^2}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac{2 \sqrt{d g-c h} \sqrt{f g-e h} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{2 d \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} b}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}}-\frac{2 \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B f g-B e h) c+2 B d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
Warning: Unable to verify antiderivative.
[In] Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(7/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
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Mathematica [B] time = 37.5099, size = 10645, normalized size = 9.44 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
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Maple [B] time = 1.96, size = 75992, normalized size = 67.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac{7}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{C b x - C a + B b}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),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((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(7/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac{7}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")
[Out]