3.105 \(\int \frac{\sqrt{c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=597 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{3 b^2 \sqrt{a+b x} (b e-a f)^2}+\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} (b e-a f)}+\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c \left (-2 A f^2+3 B e f+3 C e^2\right )+A d e f\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]

[Out]

(-2*(4*a^2*C*f + b^2*(3*B*e - 2*A*f) - a*b*(6*C*e + B*f))*Sqrt[c + d*x]*Sqrt[e +
f*x])/(3*b^2*(b*e - a*f)^2*Sqrt[a + b*x]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*x
)^(3/2)*Sqrt[e + f*x])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)) + (2*Sqrt[d
]*(8*a^3*C*d*f^2 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2*B*d*f) + a*b^2*(3*C*e*(d*e +
4*c*f) + f*(4*B*d*e + B*c*f - A*d*f)) - b^3*(A*d*e*f + c*(3*C*e^2 + 3*B*e*f - 2*
A*f^2)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]
*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^3*Sq
rt[-(b*c) + a*d]*f*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)])
+ (2*(d*e - c*f)*(4*a^2*C*d*f + b^2*(3*c*C*e + A*d*f) - a*b*(B*d*f + 3*C*(d*e +
c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF
[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a
*f))])/(3*b^3*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[e + f*
x])

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Rubi [A]  time = 3.82747, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{3 b^2 \sqrt{a+b x} (b e-a f)^2}+\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} (b e-a f)}+\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (8 a^3 C d f^2-a^2 b f (2 B d f+7 c C f+13 C d e)+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c f (3 B e-2 A f)+A d e f+3 c C e^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x]),x]

[Out]

(-2*(4*a^2*C*f + b^2*(3*B*e - 2*A*f) - a*b*(6*C*e + B*f))*Sqrt[c + d*x]*Sqrt[e +
f*x])/(3*b^2*(b*e - a*f)^2*Sqrt[a + b*x]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*x
)^(3/2)*Sqrt[e + f*x])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)) + (2*Sqrt[d
]*(8*a^3*C*d*f^2 - a^2*b*f*(13*C*d*e + 7*c*C*f + 2*B*d*f) - b^3*(3*c*C*e^2 + A*d
*e*f + c*f*(3*B*e - 2*A*f)) + a*b^2*(3*C*e*(d*e + 4*c*f) + f*(4*B*d*e + B*c*f -
A*d*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]
*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^3*Sq
rt[-(b*c) + a*d]*f*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)])
+ (2*(d*e - c*f)*(4*a^2*C*d*f + b^2*(3*c*C*e + A*d*f) - a*b*(B*d*f + 3*C*(d*e +
c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF
[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a
*f))])/(3*b^3*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[e + f*
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*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(5/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 16.8165, size = 5074, normalized size = 8.5 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(5/2)*Sqrt[e + f*x]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.115, size = 13614, normalized size = 22.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(5/2)/(f*x+e)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a} \sqrt{f x + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(d*x + c)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a
)*sqrt(f*x + e)), 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*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(5/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(5/2)*sqrt(f*x + e)), x)