### 3.1400 $$\int \frac{(c e+d e x)^{11/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx$$

Optimal. Leaf size=170 $\frac{30 e^{11/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right ),-1\right )}{77 d}-\frac{30 e^5 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{77 d}-\frac{18 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d}$

[Out]

(-30*e^5*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(77*d) - (18*e^3*(c*e + d*e*x)^(5/2)*Sqrt[1 - c^
2 - 2*c*d*x - d^2*x^2])/(77*d) - (2*e*(c*e + d*e*x)^(9/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(11*d) + (30*e^(1
1/2)*EllipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(77*d)

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Rubi [A]  time = 0.145616, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.081, Rules used = {692, 689, 221} $-\frac{30 e^5 \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{77 d}-\frac{18 e^3 \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}}{77 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{9/2}}{11 d}+\frac{30 e^{11/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*e + d*e*x)^(11/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(-30*e^5*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(77*d) - (18*e^3*(c*e + d*e*x)^(5/2)*Sqrt[1 - c^
2 - 2*c*d*x - d^2*x^2])/(77*d) - (2*e*(c*e + d*e*x)^(9/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(11*d) + (30*e^(1
1/2)*EllipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(77*d)

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 689

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(4*Sqrt[-(c/(b^2 -
4*a*c))])/e, Subst[Int[1/Sqrt[Simp[1 - (b^2*x^4)/(d^2*(b^2 - 4*a*c)), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^{11/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{1}{11} \left (9 e^2\right ) \int \frac{(c e+d e x)^{7/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{1}{77} \left (45 e^4\right ) \int \frac{(c e+d e x)^{3/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac{30 e^5 \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{1}{77} \left (15 e^6\right ) \int \frac{1}{\sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\\ &=-\frac{30 e^5 \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{\left (30 e^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{77 d}\\ &=-\frac{30 e^5 \sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{18 e^3 (c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{77 d}-\frac{2 e (c e+d e x)^{9/2} \sqrt{1-c^2-2 c d x-d^2 x^2}}{11 d}+\frac{30 e^{11/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{77 d}\\ \end{align*}

Mathematica [C]  time = 0.101802, size = 125, normalized size = 0.74 $-\frac{2 e^5 \sqrt{e (c+d x)} \left (\sqrt{-c^2-2 c d x-d^2 x^2+1} \left (c^2 \left (42 d^2 x^2+9\right )+28 c^3 d x+7 c^4+2 c d x \left (14 d^2 x^2+9\right )+7 d^4 x^4+9 d^2 x^2+15\right )-15 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right )\right )}{77 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*e + d*e*x)^(11/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(-2*e^5*Sqrt[e*(c + d*x)]*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]*(15 + 7*c^4 + 28*c^3*d*x + 9*d^2*x^2 + 7*d^4*x^4
+ 2*c*d*x*(9 + 14*d^2*x^2) + c^2*(9 + 42*d^2*x^2)) - 15*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(77*d)

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Maple [B]  time = 0.384, size = 641, normalized size = 3.8 \begin{align*} -{\frac{{e}^{5}}{2310\,d \left ({x}^{3}{d}^{3}+3\,{x}^{2}c{d}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( -900\,c+120\,{c}^{5}+924\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){c}^{3}-924\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) c+924\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) c-900\,dx+2940\,{x}^{6}c{d}^{6}+8820\,{x}^{5}{c}^{2}{d}^{5}+14700\,{x}^{4}{c}^{3}{d}^{4}+14700\,{x}^{3}{c}^{4}{d}^{3}+600\,{x}^{4}c{d}^{4}+8820\,{x}^{2}{c}^{5}{d}^{2}+1200\,{x}^{3}{c}^{2}{d}^{3}+2940\,x{c}^{6}d+1200\,{x}^{2}{c}^{3}{d}^{2}+600\,x{c}^{4}d+1080\,{x}^{2}c{d}^{2}+420\,{c}^{7}+65\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) +1155\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) -385\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) -1155\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) +360\,{c}^{3}+1080\,x{c}^{2}d-924\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){c}^{3}+420\,{x}^{7}{d}^{7}+120\,{x}^{5}{d}^{5}+360\,{x}^{3}{d}^{3} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

-1/2310*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e^5*(-900*c+120*c^5+924*(2*d*x+2*c+2)^(1/2)*(-2*d*x-2
*c+2)^(1/2)*(d*x+c)^(1/2)*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*c^3-924*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/
2)*(-2*d*x-2*c+2)^(1/2)*EllipticE(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))*c+924*(2*d*x+2*c+2)^(1/2)*(-2*d*x-2*c+2)^(1
/2)*(d*x+c)^(1/2)*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*c-900*d*x+2940*x^6*c*d^6+8820*x^5*c^2*d^5+14700*
x^4*c^3*d^4+14700*x^3*c^4*d^3+600*x^4*c*d^4+8820*x^2*c^5*d^2+1200*x^3*c^2*d^3+2940*x*c^6*d+1200*x^2*c^3*d^2+60
0*x*c^4*d+1080*x^2*c*d^2+420*c^7+65*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*(-2*d*x-2*c+2)^(1/2)*EllipticF(1/2*(2*d
*x+2*c+2)^(1/2),2^(1/2))+1155*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*(-2*d*x-2*c+2)^(1/2)*EllipticE(1/2*(2*d*x+2*c
+2)^(1/2),2^(1/2))-385*(2*d*x+2*c+2)^(1/2)*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*EllipticF(1/2*(-2*d*x-2*c+2)^(1/
2),2^(1/2))-1155*(2*d*x+2*c+2)^(1/2)*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(
1/2))+360*c^3+1080*x*c^2*d-924*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*(-2*d*x-2*c+2)^(1/2)*EllipticE(1/2*(2*d*x+2*
c+2)^(1/2),2^(1/2))*c^3+420*x^7*d^7+120*x^5*d^5+360*x^3*d^3)/d/(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3-d*x-c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{\frac{11}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^(11/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (d^{5} e^{5} x^{5} + 5 \, c d^{4} e^{5} x^{4} + 10 \, c^{2} d^{3} e^{5} x^{3} + 10 \, c^{3} d^{2} e^{5} x^{2} + 5 \, c^{4} d e^{5} x + c^{5} e^{5}\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt{d e x + c e}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-(d^5*e^5*x^5 + 5*c*d^4*e^5*x^4 + 10*c^2*d^3*e^5*x^3 + 10*c^3*d^2*e^5*x^2 + 5*c^4*d*e^5*x + c^5*e^5)*
sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*e*x + c*e)/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(11/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{\frac{11}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(11/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(11/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1), x)