3.1351 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx\)
Optimal. Leaf size=357 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}} \]
[Out]
-Sqrt[a + b*x + c*x^2]/(156*c^3*d^5*(b*d + 2*c*d*x)^(5/2)) + Sqrt[a + b*x + c*x^
2]/(78*c^3*(b^2 - 4*a*c)*d^7*Sqrt[b*d + 2*c*d*x]) - (5*(a + b*x + c*x^2)^(3/2))/
(234*c^2*d^3*(b*d + 2*c*d*x)^(9/2)) - (a + b*x + c*x^2)^(5/2)/(13*c*d*(b*d + 2*c
*d*x)^(13/2)) - (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[S
qrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(156*c^4*(b^2 - 4*a*c)^(
1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2]) + (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*
c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(
156*c^4*(b^2 - 4*a*c)^(1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2])
_______________________________________________________________________________________
Rubi [A] time = 1.05423, antiderivative size = 357, 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
\[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(15/2),x]
[Out]
-Sqrt[a + b*x + c*x^2]/(156*c^3*d^5*(b*d + 2*c*d*x)^(5/2)) + Sqrt[a + b*x + c*x^
2]/(78*c^3*(b^2 - 4*a*c)*d^7*Sqrt[b*d + 2*c*d*x]) - (5*(a + b*x + c*x^2)^(3/2))/
(234*c^2*d^3*(b*d + 2*c*d*x)^(9/2)) - (a + b*x + c*x^2)^(5/2)/(13*c*d*(b*d + 2*c
*d*x)^(13/2)) - (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[S
qrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(156*c^4*(b^2 - 4*a*c)^(
1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2]) + (Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*
c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(
156*c^4*(b^2 - 4*a*c)^(1/4)*d^(15/2)*Sqrt[a + b*x + c*x^2])
_______________________________________________________________________________________
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)**(5/2)/(2*c*d*x+b*d)**(15/2),x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [C] time = 1.90091, size = 254, normalized size = 0.71 \[ \frac{-\frac{(b+2 c x) (a+x (b+c x)) \left (31 \left (b^2-4 a c\right ) (b+2 c x)^4-28 \left (b^2-4 a c\right )^2 (b+2 c x)^2+9 \left (b^2-4 a c\right )^3-24 (b+2 c x)^6\right )}{12 c^3 \left (b^2-4 a c\right )}-\frac{i (b+2 c x)^7 \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c^4}}{156 \sqrt{a+x (b+c x)} (d (b+2 c x))^{15/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(15/2),x]
[Out]
(-((b + 2*c*x)*(a + x*(b + c*x))*(9*(b^2 - 4*a*c)^3 - 28*(b^2 - 4*a*c)^2*(b + 2*
c*x)^2 + 31*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 24*(b + 2*c*x)^6))/(12*c^3*(b^2 - 4*a*
c)) - (I*(b + 2*c*x)^7*Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]*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)/Sqrt[b^2 - 4*a*c])]], -1]))/
c^4)/(156*(d*(b + 2*c*x))^(15/2)*Sqrt[a + x*(b + c*x)])
_______________________________________________________________________________________
Maple [B] time = 0.089, size = 2125, normalized size = 6. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(15/2),x)
[Out]
1/936*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(-1760*x^6*a*c^7-1888*x^4*a^2*c^6-
1184*x^2*a^3*c^5-4056*x^5*b^3*c^5-82*x^2*b^6*c^2-1828*x^4*b^4*c^4-6*x*b^7*c-3072
*x^7*b*c^7-4936*x^6*b^2*c^6-480*x^3*b^5*c^3-480*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))*x^3*b^5*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)-180*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))*x^2*b^6*c^2
*((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)-36*Ellip
ticE(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)
)*x*b^7*c*((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)
-768*x^8*c^8+768*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))*x^6*a*c^7*((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)-192*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))*x^6*b^2*c^6*((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)-576*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))*x^5*b^3*c^5*((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)-720*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))*x^4*b^4*c^4*((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)-8*a^3*b^2*c^3
-4*a^2*b^4*c^2-6*a*b^6*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^8+12*((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^6*c+1920*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))*x^3*a*b^3*c^4*((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)+720*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))*x^2*a*b^4*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)+144*Ellip
ticE(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)
)*x*a*b^5*c^2*((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)+2304*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))*x^5*a*b*c^6*((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)+2880*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))*x^4*a*b^2*c^5*((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)-288*a^4*c^4-5280*x^5*a*b*c^6-5656*x^4*a*b^2
*c^5-3776*x^3*a^2*b*c^5-2512*x^3*a*b^3*c^4-1944*x^2*a^2*b^2*c^4-456*x^2*a*b^4*c^
3-1184*x*a^3*b*c^4-56*x*a^2*b^3*c^3-80*x*a*b^5*c^2)/d^8/(2*c^2*x^3+3*b*c*x^2+2*a
*c*x+b^2*x+a*b)/(2*c*x+b)^6/(4*a*c-b^2)/c^4
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(15/2),x, algorithm="maxima")
[Out]
integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(15/2), x)
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (128 \, c^{7} d^{7} x^{7} + 448 \, b c^{6} d^{7} x^{6} + 672 \, b^{2} c^{5} d^{7} x^{5} + 560 \, b^{3} c^{4} d^{7} x^{4} + 280 \, b^{4} c^{3} d^{7} x^{3} + 84 \, b^{5} c^{2} d^{7} x^{2} + 14 \, b^{6} c d^{7} x + b^{7} d^{7}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(15/2),x, algorithm="fricas")
[Out]
integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 +
b*x + a)/((128*c^7*d^7*x^7 + 448*b*c^6*d^7*x^6 + 672*b^2*c^5*d^7*x^5 + 560*b^3*c
^4*d^7*x^4 + 280*b^4*c^3*d^7*x^3 + 84*b^5*c^2*d^7*x^2 + 14*b^6*c*d^7*x + b^7*d^7
)*sqrt(2*c*d*x + b*d)), x)
_______________________________________________________________________________________
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)**(5/2)/(2*c*d*x+b*d)**(15/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(15/2),x, algorithm="giac")
[Out]
integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(15/2), x)