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

Optimal. Leaf size=838 \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{5/2}}{7 b d f}-\frac{2 (2 a C d f-b (7 B d f-6 C (d e+c f))) \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{3/2}}{35 b d^2 f^2}-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f)))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{a+b x}}{105 b d^3 f^3}-\frac{2 \sqrt{a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 b c e+a d e+a c f) (2 a C d f-b (7 B d f-6 C (d e+c f))))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f))))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (b e-a f) \left (-\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{e+f x}} \]

[Out]

(-2*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) + (3*a*d*f - 4*b*(d*e +
c*f))*(2*a*C*d*f - b*(7*B*d*f - 6*C*(d*e + c*f))))*Sqrt[a + b*x]*Sqrt[c + d*x]*
Sqrt[e + f*x])/(105*b*d^3*f^3) - (2*(2*a*C*d*f - b*(7*B*d*f - 6*C*(d*e + c*f)))*
(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(35*b*d^2*f^2) + (2*C*(a + b*x)^(5/
2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*a*
d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) - (3*b*c*e + a*d*e + a*c*f)*(2*a
*C*d*f - b*(7*B*d*f - 6*C*(d*e + c*f)))) + 2*((a*d*f)/2 - b*(d*e + c*f))*(5*b*d*
f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) + (3*a*d*f - 4*b*(d*e + c*f))*(2*a
*C*d*f - b*(7*B*d*f - 6*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^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b
*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3
*a*b*d*f*(7*d*f*(3*B*d*e + 2*B*c*f - 5*A*d*f) - C*(16*d^2*e^2 + 8*c*d*e*f + 11*c
^2*f^2)) - b^2*(C*(48*d^3*e^3 + 16*c*d^2*e^2*f + 17*c^2*d*e*f^2 + 24*c^3*f^3) +
7*d*f*(5*A*d*f*(2*d*e + c*f) - B*(8*d^2*e^2 + 3*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^2
*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi [A]  time = 7.85691, antiderivative size = 831, 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 \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{5/2}}{7 b d f}+\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{3/2}}{35 b d^2 f^2}-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{a+b x}}{105 b d^3 f^3}-\frac{2 \sqrt{a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (b e-a f) \left (-\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) - (3*a*d*f - 4*b*(d*e +
c*f))*(7*b*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*
Sqrt[e + f*x])/(105*b*d^3*f^3) + (2*(7*b*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f))*
(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(35*b*d^2*f^2) + (2*C*(a + b*x)^(5/
2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*a*
d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) + (3*b*c*e + a*d*e + a*c*f)*(7*b
*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f))) + 2*((a*d*f)/2 - b*(d*e + c*f))*(5*b*d*
f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) - (3*a*d*f - 4*b*(d*e + c*f))*(7*b
*B*d*f - 2*a*C*d*f - 6*b*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^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b
*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3
*a*b*d*f*(7*d*f*(3*B*d*e + 2*B*c*f - 5*A*d*f) - C*(16*d^2*e^2 + 8*c*d*e*f + 11*c
^2*f^2)) - b^2*(C*(48*d^3*e^3 + 16*c*d^2*e^2*f + 17*c^2*d*e*f^2 + 24*c^3*f^3) +
7*d*f*(5*A*d*f*(2*d*e + c*f) - B*(8*d^2*e^2 + 3*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^2
*d^(7/2)*f^4*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((b*x+a)**(3/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 19.9034, size = 7300, normalized size = 8.71 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.082, size = 10546, normalized size = 12.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(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. \[ \int \frac{{\left (C x^{2} + B x + A\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{\sqrt{d x + c} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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