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

Optimal. Leaf size=774 \[ -\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)+b^2 \left (-\left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))-(3 a c f+a d e+b c e) (6 a C d f-b (7 B d f-4 C (c f+d e))))+2 \left (\frac{b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))-(4 a d f-b c f+2 b d e) (6 a C d f-b (7 B d f-4 C (c f+d e))))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))-(4 a d f-b c f+2 b d e) (6 a C d f-b (7 B d f-4 C (c f+d e))))}{105 b^3 d^2 f^2}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} (6 a C d f-b (7 B d f-4 C (c f+d e)))}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f} \]

[Out]

(-2*(5*b*d*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f)) - (2*b*d*e - b*c*f + 4*a*
d*f)*(6*a*C*d*f - b*(7*B*d*f - 4*C*(d*e + c*f))))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sq
rt[e + f*x])/(105*b^3*d^2*f^2) - (2*(6*a*C*d*f - b*(7*B*d*f - 4*C*(d*e + c*f)))*
Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(35*b^2*d*f^2) + (2*C*Sqrt[a + b*x]
*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*
b*c*f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f)) - (b*c*e + a*d*e + 3*a*c*f)*(6*a
*C*d*f - b*(7*B*d*f - 4*C*(d*e + c*f)))) + 2*((b*d*e)/2 - (b*c + a*d)*f)*(5*b*d*
f*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f)) - (2*b*d*e - b*c*f + 4*a*d*f)*(6*a*C
*d*f - b*(7*B*d*f - 4*C*(d*e + c*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))])/(105*b^4*d^(5/2)*f^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e
- a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(24*a^2*C*d^2*f^2 + a*
b*d*f*(13*C*d*e - 5*c*C*f - 28*B*d*f) - b^2*(7*d*f*(2*B*d*e - B*c*f - 5*A*d*f) -
C*(8*d^2*e^2 - c*d*e*f - 4*c^2*f^2)))*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))])/(105*b^4*d^(5/2)*f^3*Sqrt[c + d*x]*Sqrt[e
+ f*x])

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Rubi [A]  time = 6.89813, antiderivative size = 769, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184 \[ -\frac{2 \sqrt{a d-b c} (b e-a f) (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (24 a^2 C d^2 f^2+a b d f (-28 B d f-5 c C f+13 C d e)+b^2 \left (-\left (7 d f (-5 A d f-B c f+2 B d e)-C \left (-4 c^2 f^2-c d e f+8 d^2 e^2\right )\right )\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (3 b d f (5 b c f (3 a C (c f+d e)+b (c C e-7 A d f))+(3 a c f+a d e+b c e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))+2 \left (\frac{b d e}{2}-f (a d+b c)\right ) (5 b d f (3 a C (c f+d e)+b (c C e-7 A d f))+(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e)))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^4 d^{5/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \left (5 (3 a C (c f+d e)+b (c C e-7 A d f))+\frac{(4 a d f-b c f+2 b d e) (-6 a C d f+7 b B d f-4 b C (c f+d e))}{b d f}\right )}{105 b^2 d f}+\frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^{3/2} (-6 a C d f+7 b B d f-4 b C (c f+d e))}{35 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} (e+f x)^{3/2}}{7 b d f} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(((2*b*d*e - b*c*f + 4*a*d*f)*(7*b*B*d*f - 6*a*C*d*f - 4*b*C*(d*e + c*f)))/(
b*d*f) + 5*(3*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f)))*Sqrt[a + b*x]*Sqrt[c + d*x
]*Sqrt[e + f*x])/(105*b^2*d*f) + (2*(7*b*B*d*f - 6*a*C*d*f - 4*b*C*(d*e + c*f))*
Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2))/(35*b^2*d*f^2) + (2*C*Sqrt[a + b*x]
*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*((b
*c*e + a*d*e + 3*a*c*f)*(7*b*B*d*f - 6*a*C*d*f - 4*b*C*(d*e + c*f)) + 5*b*c*f*(3
*a*C*(d*e + c*f) + b*(c*C*e - 7*A*d*f))) + 2*((b*d*e)/2 - (b*c + a*d)*f)*((2*b*d
*e - b*c*f + 4*a*d*f)*(7*b*B*d*f - 6*a*C*d*f - 4*b*C*(d*e + c*f)) + 5*b*d*f*(3*a
*C*(d*e + c*f) + b*(c*C*e - 7*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))])/(105*b^4*d^(5/2)*f^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e
- a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(24*a^2*C*d^2*f^2 + a*
b*d*f*(13*C*d*e - 5*c*C*f - 28*B*d*f) - b^2*(7*d*f*(2*B*d*e - B*c*f - 5*A*d*f) -
C*(8*d^2*e^2 - c*d*e*f - 4*c^2*f^2)))*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))])/(105*b^4*d^(5/2)*f^3*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)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 18.7491, size = 7297, normalized size = 9.43 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.076, size = 10268, normalized size = 13.3 \[ \text{output too large to display} \]

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)^(1/2),x)

[Out]

result too large to display

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

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)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

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

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

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)/sqrt(b*x + a),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

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)/sqrt(b*x + a),x, algorithm="giac")

[Out]

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