3.122 \(\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

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]

(2*b*d*(9*a^3*C*d*f*h - b^3*(2*B*d*e*g - c*(3*C*e*g - 2*B*f*g - 2*B*e*h)) + a*b^
2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) - a^2*b*(6*B*d*f*h +
5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*
c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[c + d*x]) - (2*b^2*(b*B - 2*a*C)*Sqr
t[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(
a + b*x)^(3/2)) - (2*b^2*(9*a^3*C*d*f*h - b^3*(2*B*d*e*g - c*(3*C*e*g - 2*B*f*g
- 2*B*e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) -
a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqr
t[g + h*x])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[a + b*x]) - (2*b*S
qrt[d*g - c*h]*Sqrt[f*g - e*h]*(9*a^3*C*d*f*h - b^3*(2*B*d*e*g - c*(3*C*e*g - 2*
B*f*g - 2*B*e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*
h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[-(((d*
e - c*f)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*
Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e
- a*f)*(d*g - c*h))])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[((d*e -
c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*(3*a^3*C*d^2*f*h -
b^3*(2*B*d^2*e*g - B*c^2*f*h - c*d*(3*C*e*g - B*f*g - B*e*h)) - 3*a^2*b*d*(B*d*
f*h + C*(d*f*g + d*e*h - c*f*h)) + a*b^2*(3*B*d^2*(f*g + e*h) + C*(d^2*e*g - c*d
*f*g - c*d*e*h - 2*c^2*f*h)))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x
))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e
*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/(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[
-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])

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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]

(2*b*d*(9*a^3*C*d*f*h + b^3*(3*c*C*e*g - 2*B*d*e*g - 2*B*c*(f*g + e*h)) + a*b^2*
(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) - a^2*b*(6*B*d*f*h + 5
*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c
- a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[c + d*x]) - (2*b^2*(b*B - 2*a*C)*Sqrt[
c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a
+ b*x)^(3/2)) - (2*b^2*(9*a^3*C*d*f*h + b^3*(3*c*C*e*g - 2*B*d*e*g - 2*B*c*(f*g
+ e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) - a^2*
b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g
+ h*x])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[a + b*x]) - (2*b*Sqrt[
d*g - c*h]*Sqrt[f*g - e*h]*(9*a^3*C*d*f*h + b^3*(3*c*C*e*g - 2*B*d*e*g - 2*B*c*(
f*g + e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) -
a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*
f)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e
+ f*x])/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f
)*(d*g - c*h))])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g - a*h)^2*Sqrt[((d*e - c*f)*
(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*(3*a^3*C*d^2*f*h - b^3*(
2*B*d^2*e*g - B*c^2*f*h - c*d*(3*C*e*g - B*f*g - B*e*h)) - 3*a^2*b*d*(B*d*f*h +
C*(d*f*g + d*e*h - c*f*h)) + a*b^2*(3*B*d^2*(f*g + e*h) + C*(d^2*e*g - c*d*f*g -
c*d*e*h - 2*c^2*f*h)))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sq
rt[g + h*x]*EllipticF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sq
rt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/(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[-(((b*
e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])

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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]

Timed 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]

Result too large to show

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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]

result too large to display

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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]

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)*s
qrt(f*x + e)*sqrt(h*x + g)), x)

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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]

integral((C*b*x - C*a + B*b)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*sqrt(d*x +
c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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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]

Timed 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]

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)*s
qrt(f*x + e)*sqrt(h*x + g)), x)