3.97 \(\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=706 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} \left (6 a^2 C d f-a b (5 B d f+c C f+C d e)+b^2 (5 A d f+c C e)\right )}{5 b^2 f (b c-a d) (b e-a f)}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{15 b^3 d f (b e-a f)}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)} \]

[Out]

(2*(24*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B*d*f) + b^2*(5*d*f*(B*e + 3*A*
f) - C*e*(2*d*e - c*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b^3*d*f*
(b*e - a*f)) + (2*(6*a^2*C*d*f + b^2*(c*C*e + 5*A*d*f) - a*b*(C*d*e + c*C*f + 5*
B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(5*b^2*(b*c - a*d)*f*(b*e -
a*f)) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a
*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt[-(b*c) + a*d]*(48*a^2*C*d^2*f^2 - 8*a*b
*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f) - 2*C*(d^2
*e^2 - c*d*e*f + c^2*f^2)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Ellipt
icE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e
- a*f))])/(15*b^4*d^(3/2)*f^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (
2*Sqrt[-(b*c) + a*d]*(d*e - c*f)*(24*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B
*d*f) + b^2*(5*d*f*(B*e + 3*A*f) - C*e*(2*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))])/(15*b^4*d^(3/2)*f^2*Sqrt[
c + d*x]*Sqrt[e + f*x])

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Rubi [A]  time = 5.12922, antiderivative size = 706, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} \left (6 a^2 C d f-a b (5 B d f+c C f+C d e)+b^2 (5 A d f+c C e)\right )}{5 b^2 f (b c-a d) (b e-a f)}+\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{15 b^3 d f (b e-a f)}-\frac{2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(3/2),x]

[Out]

(2*(24*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B*d*f) + b^2*(5*d*f*(B*e + 3*A*
f) - C*e*(2*d*e - c*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b^3*d*f*
(b*e - a*f)) + (2*(6*a^2*C*d*f + b^2*(c*C*e + 5*A*d*f) - a*b*(C*d*e + c*C*f + 5*
B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(5*b^2*(b*c - a*d)*f*(b*e -
a*f)) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a
*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt[-(b*c) + a*d]*(48*a^2*C*d^2*f^2 - 8*a*b
*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f) - 2*C*(d^2
*e^2 - c*d*e*f + c^2*f^2)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Ellipt
icE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e
- a*f))])/(15*b^4*d^(3/2)*f^2*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (
2*Sqrt[-(b*c) + a*d]*(d*e - c*f)*(24*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B
*d*f) + b^2*(5*d*f*(B*e + 3*A*f) - C*e*(2*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))])/(15*b^4*d^(3/2)*f^2*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)*(f*x+e)**(1/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 15.6284, size = 9487, normalized size = 13.44 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x)^(3/2),x]

[Out]

Result too large to show

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Maple [B]  time = 0.095, size = 6257, normalized size = 8.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(3/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c} \sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c} \sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(3/2),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(3/2), 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)*(f*x+e)**(1/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c} \sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(3/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a)^(3/2), x)