3.1569 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=627 \[ \frac{\left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac{\sqrt{a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{256 c^2 e^6}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{96 c e^4}-\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^7}-\frac{\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2} \]

[Out]

-((512*c^5*d^5 + b^5*e^5 - 128*c^4*d^3*e*(11*b*d - 8*a*e) + 8*b^3*c*e^4*(b*d - 2
*a*e) + 32*c^3*d*e^2*(40*b^2*d^2 - 55*a*b*d*e + 16*a^2*e^2) - 8*b*c^2*e^3*(49*b^
2*d^2 - 92*a*b*d*e + 42*a^2*e^2) - 2*c*e*(16*c^2*d^2 - b^2*e^2 - 4*c*e*(4*b*d -
5*a*e))*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*Sqrt[a + b*x + c*x^2])/(2
56*c^2*e^6) - ((64*c^3*d^3 - b^3*e^3 + 4*b*c*e^2*(14*b*d - 13*a*e) - 8*c^2*d*e*(
15*b*d - 8*a*e) - 2*c*e*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x)*(a + b
*x + c*x^2)^(3/2))/(96*c*e^4) - ((12*c*d - 11*b*e - 10*c*e*x)*(a + b*x + c*x^2)^
(5/2))/(30*e^2) + ((1024*c^6*d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c
^5*d^4*e*(6*b*d - 5*a*e) - 320*c^3*e^3*(b*d - a*e)^2*(4*b*d - a*e) + 640*c^4*d^2
*e^2*(5*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + 40*b^2*c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e
+ 6*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(5
/2)*e^7) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTanh[(b*d - 2*a*e + (
2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e^7

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Rubi [A]  time = 2.51007, antiderivative size = 627, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)-320 c^3 e^3 (b d-a e)^2 (4 b d-a e)+b^6 e^6+1024 c^6 d^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{5/2} e^7}-\frac{\sqrt{a+b x+c x^2} \left (32 c^3 d e^2 \left (16 a^2 e^2-55 a b d e+40 b^2 d^2\right )-8 b c^2 e^3 \left (42 a^2 e^2-92 a b d e+49 b^2 d^2\right )+8 b^3 c e^4 (b d-2 a e)-2 c e x \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-128 c^4 d^3 e (11 b d-8 a e)+b^5 e^5+512 c^5 d^5\right )}{256 c^2 e^6}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-8 c^2 d e (15 b d-8 a e)+4 b c e^2 (14 b d-13 a e)-b^3 e^3+64 c^3 d^3\right )}{96 c e^4}-\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^7}-\frac{\left (a+b x+c x^2\right )^{5/2} (-11 b e+12 c d-10 c e x)}{30 e^2} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x),x]

[Out]

-((512*c^5*d^5 + b^5*e^5 - 128*c^4*d^3*e*(11*b*d - 8*a*e) + 8*b^3*c*e^4*(b*d - 2
*a*e) + 32*c^3*d*e^2*(40*b^2*d^2 - 55*a*b*d*e + 16*a^2*e^2) - 8*b*c^2*e^3*(49*b^
2*d^2 - 92*a*b*d*e + 42*a^2*e^2) - 2*c*e*(16*c^2*d^2 - b^2*e^2 - 4*c*e*(4*b*d -
5*a*e))*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*Sqrt[a + b*x + c*x^2])/(2
56*c^2*e^6) - ((64*c^3*d^3 - b^3*e^3 + 4*b*c*e^2*(14*b*d - 13*a*e) - 8*c^2*d*e*(
15*b*d - 8*a*e) - 2*c*e*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x)*(a + b
*x + c*x^2)^(3/2))/(96*c*e^4) - ((12*c*d - 11*b*e - 10*c*e*x)*(a + b*x + c*x^2)^
(5/2))/(30*e^2) + ((1024*c^6*d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c
^5*d^4*e*(6*b*d - 5*a*e) - 320*c^3*e^3*(b*d - a*e)^2*(4*b*d - a*e) + 640*c^4*d^2
*e^2*(5*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + 40*b^2*c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e
+ 6*a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(5
/2)*e^7) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(5/2)*ArcTanh[(b*d - 2*a*e + (
2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e^7

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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((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d),x)

[Out]

Timed out

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Mathematica [A]  time = 2.00752, size = 637, normalized size = 1.02 \[ \frac{\frac{15 \left (640 c^4 d^2 e^2 \left (3 a^2 e^2-8 a b d e+5 b^2 d^2\right )+40 b^2 c^2 e^4 \left (6 a^2 e^2-8 a b d e+3 b^2 d^2\right )+4 b^4 c e^5 (2 b d-5 a e)-512 c^5 d^4 e (6 b d-5 a e)+320 c^3 e^3 (b d-a e)^2 (a e-4 b d)+b^6 e^6+1024 c^6 d^6\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{5/2}}+\frac{2 e \sqrt{a+x (b+c x)} \left (-16 c^3 e^2 \left (2 a^2 e^2 (368 d-165 e x)-2 a b e \left (975 d^2-446 d e x+283 e^2 x^2\right )+b^2 \left (1200 d^3-555 d^2 e x+356 d e^2 x^2-261 e^3 x^3\right )\right )+8 b c^2 e^3 \left (1066 a^2 e^2+2 a b e (341 e x-830 d)+b^2 \left (735 d^2-310 d e x+191 e^2 x^2\right )\right )+10 b^3 c e^4 (28 a e-12 b d+b e x)+64 c^4 e \left (a e \left (-280 d^3+135 d^2 e x-88 d e^2 x^2+65 e^3 x^3\right )+b \left (330 d^4-160 d^3 e x+105 d^2 e^2 x^2-78 d e^3 x^3+62 e^4 x^4\right )\right )-15 b^5 e^5-128 c^5 \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )}{c^2}-7680 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^{5/2}+7680 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^{5/2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{7680 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x),x]

[Out]

((2*e*Sqrt[a + x*(b + c*x)]*(-15*b^5*e^5 + 10*b^3*c*e^4*(-12*b*d + 28*a*e + b*e*
x) - 128*c^5*(60*d^5 - 30*d^4*e*x + 20*d^3*e^2*x^2 - 15*d^2*e^3*x^3 + 12*d*e^4*x
^4 - 10*e^5*x^5) + 8*b*c^2*e^3*(1066*a^2*e^2 + 2*a*b*e*(-830*d + 341*e*x) + b^2*
(735*d^2 - 310*d*e*x + 191*e^2*x^2)) - 16*c^3*e^2*(2*a^2*e^2*(368*d - 165*e*x) -
2*a*b*e*(975*d^2 - 446*d*e*x + 283*e^2*x^2) + b^2*(1200*d^3 - 555*d^2*e*x + 356
*d*e^2*x^2 - 261*e^3*x^3)) + 64*c^4*e*(a*e*(-280*d^3 + 135*d^2*e*x - 88*d*e^2*x^
2 + 65*e^3*x^3) + b*(330*d^4 - 160*d^3*e*x + 105*d^2*e^2*x^2 - 78*d*e^3*x^3 + 62
*e^4*x^4))))/c^2 - 7680*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*Log[d + e
*x] + (15*(1024*c^6*d^6 + b^6*e^6 + 4*b^4*c*e^5*(2*b*d - 5*a*e) - 512*c^5*d^4*e*
(6*b*d - 5*a*e) + 320*c^3*e^3*(b*d - a*e)^2*(-4*b*d + a*e) + 640*c^4*d^2*e^2*(5*
b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2) + 40*b^2*c^2*e^4*(3*b^2*d^2 - 8*a*b*d*e + 6*a^2
*e^2))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2) + 7680*(2*c*d -
b*e)*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*
Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(7680*e^7)

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Maple [B]  time = 0.018, size = 6077, normalized size = 9.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [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)^(5/2)*(2*c*x + b)/(e*x + d),x, algorithm="fricas")

[Out]

Timed out

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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((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError