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

Optimal. Leaf size=1235 \[ \text{result too large to display} \]

[Out]

(-2*(5*b*d*f*(7*a*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (3*b*c*e +
3*a*d*e + a*c*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))) + 2*((a*d*f)/2
- b*(2*d*e + c*f))*(7*b*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (6*b
*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))))*Sqrt[a
+ b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(945*b^2*d^3*f^4) - (2*(7*b*d*f*(5*b*c*C*e
+ 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (6*b*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f +
b*(8*C*d*e + 6*c*C*f - 9*B*d*f)))*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/(
315*b*d^3*f^3) - (2*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))*(a + b*x)^(3/2
)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(63*b*d^2*f^2) + (2*C*(a + b*x)^(5/2)*(c + d*x)
^(3/2)*Sqrt[e + f*x])/(9*b*d*f) + (2*Sqrt[-(b*c) + a*d]*(8*a^4*C*d^4*f^4 + a^3*b
*d^3*f^3*(11*C*d*e - 7*c*C*f - 18*B*d*f) - 3*a^2*b^2*d^2*f^2*(3*d*f*(4*B*d*e - 3
*B*c*f - 7*A*d*f) - C*(9*d^2*e^2 - 5*c*d*e*f - 3*c^2*f^2)) - a*b^3*d*f*(2*C*(92*
d^3*e^3 - 33*c*d^2*e^2*f - 18*c^2*d*e*f^2 - 16*c^3*f^3) + 3*d*f*(7*A*d*f*(13*d*e
- 7*c*f) - B*(72*d^2*e^2 - 29*c*d*e*f - 19*c^2*f^2))) + b^4*(C*(128*d^4*e^4 - 4
0*c*d^3*e^3*f - 21*c^2*d^2*e^2*f^2 - 16*c^3*d*e*f^3 - 16*c^4*f^4) + 3*d*f*(7*A*d
*f*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) - B*(48*d^3*e^3 - 16*c*d^2*e^2*f - 9*c^2*
d*e*f^2 - 8*c^3*f^3))))*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))])/(315*b^3*d^(7/2)*f^5*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*S
qrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(4*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(3*
C*d*e - c*C*f - 3*B*d*f) - 3*a*b^2*d*f*(3*d*f*(16*B*d*e + 3*B*c*f - 21*A*d*f) -
5*C*(8*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) - b^3*(C*(128*d^3*e^3 + 24*c*d^2*e^2*f +
15*c^2*d*e*f^2 + 8*c^3*f^3) + 3*d*f*(7*A*d*f*(8*d*e + c*f) - 4*B*(12*d^2*e^2 + 2
*c*d*e*f + 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))])/(315*b^3*d^(7/2)*f^5*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi [A]  time = 14.8257, antiderivative size = 1235, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184 \[ \frac{2 C (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^{5/2}}{9 b d f}-\frac{2 (4 a C d f+b (8 C d e+6 c C f-9 B d f)) (c+d x)^{3/2} \sqrt{e+f x} (a+b x)^{3/2}}{63 b d^2 f^2}-\frac{2 (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f))) (c+d x)^{3/2} \sqrt{e+f x} \sqrt{a+b x}}{315 b d^3 f^3}-\frac{2 \left (5 b d f (7 a d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(3 b c e+3 a d e+a c f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))+2 \left (\frac{a d f}{2}-b (2 d e+c f)\right ) (7 b d f (5 b c C e+3 a C d e+a c C f-9 A b d f)-(6 b d e+4 b c f-3 a d f) (4 a C d f+b (8 C d e+6 c C f-9 B d f)))\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{a+b x}}{945 b^2 d^3 f^4}+\frac{2 \sqrt{a d-b c} \left (\left (C \left (128 d^4 e^4-40 c d^3 f e^3-21 c^2 d^2 f^2 e^2-16 c^3 d f^3 e-16 c^4 f^4\right )+3 d f \left (7 A d f \left (8 d^2 e^2-3 c d f e-2 c^2 f^2\right )-B \left (48 d^3 e^3-16 c d^2 f e^2-9 c^2 d f^2 e-8 c^3 f^3\right )\right )\right ) b^4-a d f \left (2 C \left (92 d^3 e^3-33 c d^2 f e^2-18 c^2 d f^2 e-16 c^3 f^3\right )+3 d f \left (7 A d f (13 d e-7 c f)-B \left (72 d^2 e^2-29 c d f e-19 c^2 f^2\right )\right )\right ) b^3-3 a^2 d^2 f^2 \left (3 d f (4 B d e-3 B c f-7 A d f)-C \left (9 d^2 e^2-5 c d f e-3 c^2 f^2\right )\right ) b^2+a^3 d^3 f^3 (11 C d e-7 c C f-18 B d f) b+8 a^4 C d^4 f^4\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 )}{315 b^3 d^{7/2} f^5 \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) (d e-c f) \left (-\left (C \left (128 d^3 e^3+24 c d^2 f e^2+15 c^2 d f^2 e+8 c^3 f^3\right )+3 d f \left (7 A d f (8 d e+c f)-4 B \left (12 d^2 e^2+2 c d f e+c^2 f^2\right )\right )\right ) b^3-3 a d f \left (3 d f (16 B d e+3 B c f-21 A d f)-5 C \left (8 d^2 e^2+2 c d f e+c^2 f^2\right )\right ) b^2+3 a^2 d^2 f^2 (3 C d e-c C f-3 B d f) b+4 a^3 C d^3 f^3\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 )}{315 b^3 d^{7/2} f^5 \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(5*b*d*f*(7*a*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (3*b*c*e +
3*a*d*e + a*c*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))) + 2*((a*d*f)/2
- b*(2*d*e + c*f))*(7*b*d*f*(5*b*c*C*e + 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (6*b
*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))))*Sqrt[a
+ b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(945*b^2*d^3*f^4) - (2*(7*b*d*f*(5*b*c*C*e
+ 3*a*C*d*e + a*c*C*f - 9*A*b*d*f) - (6*b*d*e + 4*b*c*f - 3*a*d*f)*(4*a*C*d*f +
b*(8*C*d*e + 6*c*C*f - 9*B*d*f)))*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/(
315*b*d^3*f^3) - (2*(4*a*C*d*f + b*(8*C*d*e + 6*c*C*f - 9*B*d*f))*(a + b*x)^(3/2
)*(c + d*x)^(3/2)*Sqrt[e + f*x])/(63*b*d^2*f^2) + (2*C*(a + b*x)^(5/2)*(c + d*x)
^(3/2)*Sqrt[e + f*x])/(9*b*d*f) + (2*Sqrt[-(b*c) + a*d]*(8*a^4*C*d^4*f^4 + a^3*b
*d^3*f^3*(11*C*d*e - 7*c*C*f - 18*B*d*f) - 3*a^2*b^2*d^2*f^2*(3*d*f*(4*B*d*e - 3
*B*c*f - 7*A*d*f) - C*(9*d^2*e^2 - 5*c*d*e*f - 3*c^2*f^2)) - a*b^3*d*f*(2*C*(92*
d^3*e^3 - 33*c*d^2*e^2*f - 18*c^2*d*e*f^2 - 16*c^3*f^3) + 3*d*f*(7*A*d*f*(13*d*e
- 7*c*f) - B*(72*d^2*e^2 - 29*c*d*e*f - 19*c^2*f^2))) + b^4*(C*(128*d^4*e^4 - 4
0*c*d^3*e^3*f - 21*c^2*d^2*e^2*f^2 - 16*c^3*d*e*f^3 - 16*c^4*f^4) + 3*d*f*(7*A*d
*f*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2) - B*(48*d^3*e^3 - 16*c*d^2*e^2*f - 9*c^2*
d*e*f^2 - 8*c^3*f^3))))*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))])/(315*b^3*d^(7/2)*f^5*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*S
qrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(4*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(3*
C*d*e - c*C*f - 3*B*d*f) - 3*a*b^2*d*f*(3*d*f*(16*B*d*e + 3*B*c*f - 21*A*d*f) -
5*C*(8*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) - b^3*(C*(128*d^3*e^3 + 24*c*d^2*e^2*f +
15*c^2*d*e*f^2 + 8*c^3*f^3) + 3*d*f*(7*A*d*f*(8*d*e + c*f) - 4*B*(12*d^2*e^2 + 2
*c*d*e*f + 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))])/(315*b^3*d^(7/2)*f^5*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 = 23.916, size = 18421, normalized size = 14.92 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.088, size = 15857, normalized size = 12.8 \[ \text{output too large to display} \]

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)