3.1346 \(\int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=373 \[ \frac{d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{884 c^4 \sqrt{a+b x+c x^2}}-\frac{d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{884 c^4 \sqrt{a+b x+c x^2}}-\frac{d \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{1326 c^3}+\frac{5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{2652 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{442 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{7/2}}{17 c d} \]

[Out]

-((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(1326*c^3) + (5
*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(2652*c^3*d) - (5*
(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2))/(442*c^2*d) + ((b*d
+ 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2))/(17*c*d) - ((b^2 - 4*a*c)^(19/4)*d^(5
/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c
*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(884*c^4*Sqrt[a + b*x + c*x^2]) + ((b
^2 - 4*a*c)^(19/4)*d^(5/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elliptic
F[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(884*c^4*Sqrt[
a + b*x + c*x^2])

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Rubi [A]  time = 1.16354, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{884 c^4 \sqrt{a+b x+c x^2}}-\frac{d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{884 c^4 \sqrt{a+b x+c x^2}}-\frac{d \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{1326 c^3}+\frac{5 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{2652 c^3 d}-\frac{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{442 c^2 d}+\frac{\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{7/2}}{17 c d} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(1326*c^3) + (5
*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(2652*c^3*d) - (5*
(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2))/(442*c^2*d) + ((b*d
+ 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2))/(17*c*d) - ((b^2 - 4*a*c)^(19/4)*d^(5
/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c
*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(884*c^4*Sqrt[a + b*x + c*x^2]) + ((b
^2 - 4*a*c)^(19/4)*d^(5/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elliptic
F[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(884*c^4*Sqrt[
a + b*x + c*x^2])

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

[Out]

Timed out

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Mathematica [C]  time = 2.53756, size = 327, normalized size = 0.88 \[ \frac{(d (b+2 c x))^{5/2} \left (\frac{c (a+x (b+c x)) \left (4 b^2 c^2 \left (65 a^2+470 a c x^2+477 c^2 x^4\right )+16 b c^3 x \left (89 a^2+216 a c x^2+117 c^2 x^4\right )+16 c^3 \left (8 a^3+89 a^2 c x^2+108 a c^2 x^4+39 c^3 x^6\right )+b^4 \left (26 c^2 x^2-46 a c\right )+8 b^3 c^2 x \left (19 a+87 c x^2\right )+3 b^6-10 b^5 c x\right )}{b+2 c x}+\frac{3 i \left (b^2-4 a c\right )^{7/2} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{5/2}}\right )}{2652 c^4 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(5/2)*((c*(a + x*(b + c*x))*(3*b^6 - 10*b^5*c*x + 8*b^3*c^2*x*(
19*a + 87*c*x^2) + b^4*(-46*a*c + 26*c^2*x^2) + 16*b*c^3*x*(89*a^2 + 216*a*c*x^2
+ 117*c^2*x^4) + 4*b^2*c^2*(65*a^2 + 470*a*c*x^2 + 477*c^2*x^4) + 16*c^3*(8*a^3
+ 89*a^2*c*x^2 + 108*a*c^2*x^4 + 39*c^3*x^6)))/(b + 2*c*x) + ((3*I)*(b^2 - 4*a*
c)^(7/2)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(EllipticE[I*ArcSinh[Sqrt[-(
(b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/S
qrt[b^2 - 4*a*c])]], -1]))/(-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(5/2)))/(2652*c^4*
Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.039, size = 1190, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5304*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^2*(-256*a^4*b^2*c^4-520*a^3*b^
4*c^3+92*a^2*b^6*c^2-6*a*b^8*c+1920*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(
1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))
/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^3*b^4*c^3-480*((b+2*c*x+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a
*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1
/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^2*b^6*c^2+60*((b+2*c*x+(-4*a*c
+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-
b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+2*c*x+(-
4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a*b^8*c-3*((b+2*c*x
+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(
1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((b+
2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*b^10+3072*(
(b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^
(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(
1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^5
*c^5-75264*x^7*a*b*c^8-119104*x^6*a*b^2*c^7-75648*x^5*a^2*b*c^7-93888*x^5*a*b^3*
c^6-85248*x^4*a^2*b^2*c^6-37368*x^4*a*b^4*c^5-25216*x^6*a^2*c^8-12416*x^4*a^3*c^
7-1024*x^2*a^4*c^6-10*x^2*b^8*c^2-24960*x^9*b*c^9-24832*x^3*a^3*b*c^6-44416*x^3*
a^2*b^3*c^5-6064*x^3*a*b^5*c^4-17600*x^2*a^3*b^2*c^5-10056*x^2*a^2*b^4*c^4+160*x
^2*a*b^6*c^3-1024*x*a^4*b*c^5-5184*x*a^3*b^3*c^4-456*x*a^2*b^5*c^3+88*x*a*b^7*c^
2-3840*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a
*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*El
lipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1
/2))*a^4*b^2*c^4-18816*x^8*a*c^9-34168*x^6*b^4*c^6-51456*x^8*b^2*c^8-1516*x^4*b^
6*c^4-6*x*b^9*c-11112*x^5*b^5*c^5-56064*x^7*b^3*c^7-4992*x^10*c^10)/c^4/(2*c^2*x
^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (4 \, c^{4} d^{2} x^{6} + 12 \, b c^{3} d^{2} x^{5} +{\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{4} + a^{2} b^{2} d^{2} + 2 \,{\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{3} +{\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{2} + 2 \,{\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*c^4*d^2*x^6 + 12*b*c^3*d^2*x^5 + (13*b^2*c^2 + 8*a*c^3)*d^2*x^4 + a^
2*b^2*d^2 + 2*(3*b^3*c + 8*a*b*c^2)*d^2*x^3 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2
*x^2 + 2*(a*b^3 + 2*a^2*b*c)*d^2*x)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + 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((2*c*d*x+b*d)**(5/2)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError