3.17.49 \(\int \frac {1+x^2}{\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx\)

Optimal. Leaf size=111 \[ \frac {4 \left (35 a^2 x^2+315 a^2-56 a b x+48 a c^2 x+224 b^2-288 b c^2+128 c^4\right ) \sqrt {\sqrt {a x+b}+c}}{315 a^3}-\frac {32 \sqrt {a x+b} \left (5 a c x-16 b c+8 c^3\right ) \sqrt {\sqrt {a x+b}+c}}{315 a^3} \]

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Rubi [A]  time = 0.47, antiderivative size = 138, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1850} \begin {gather*} -\frac {8 \left (b-3 c^2\right ) \left (\sqrt {a x+b}+c\right )^{5/2}}{5 a^3}+\frac {16 c \left (b-c^2\right ) \left (\sqrt {a x+b}+c\right )^{3/2}}{3 a^3}+\frac {4 \left (\sqrt {a x+b}+c\right )^{9/2}}{9 a^3}-\frac {16 c \left (\sqrt {a x+b}+c\right )^{7/2}}{7 a^3}+\frac {4 \left (a^2+\left (b-c^2\right )^2\right ) \sqrt {\sqrt {a x+b}+c}}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

(4*(a^2 + (b - c^2)^2)*Sqrt[c + Sqrt[b + a*x]])/a^3 + (16*c*(b - c^2)*(c + Sqrt[b + a*x])^(3/2))/(3*a^3) - (8*
(b - 3*c^2)*(c + Sqrt[b + a*x])^(5/2))/(5*a^3) - (16*c*(c + Sqrt[b + a*x])^(7/2))/(7*a^3) + (4*(c + Sqrt[b + a
*x])^(9/2))/(9*a^3)

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {align*} \int \frac {1+x^2}{\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {a^2+\left (b-x^2\right )^2}{\sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a^3}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {a^2+b^2-2 b c^2+c^4}{\sqrt {c+x}}-4 c \left (-b+c^2\right ) \sqrt {c+x}-2 \left (b-3 c^2\right ) (c+x)^{3/2}-4 c (c+x)^{5/2}+(c+x)^{7/2}\right ) \, dx,x,\sqrt {b+a x}\right )}{a^3}\\ &=\frac {4 \left (a^2+\left (b-c^2\right )^2\right ) \sqrt {c+\sqrt {b+a x}}}{a^3}+\frac {16 c \left (b-c^2\right ) \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a^3}-\frac {8 \left (b-3 c^2\right ) \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a^3}-\frac {16 c \left (c+\sqrt {b+a x}\right )^{7/2}}{7 a^3}+\frac {4 \left (c+\sqrt {b+a x}\right )^{9/2}}{9 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 102, normalized size = 0.92 \begin {gather*} \frac {4 \sqrt {\sqrt {a x+b}+c} \left (35 a^2 \left (x^2+9\right )+32 \left (-2 c^3 \left (\sqrt {a x+b}-2 c\right )+b c \left (4 \sqrt {a x+b}-9 c\right )+7 b^2\right )-8 a x \left (c \left (5 \sqrt {a x+b}-6 c\right )+7 b\right )\right )}{315 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

(4*Sqrt[c + Sqrt[b + a*x]]*(35*a^2*(9 + x^2) + 32*(7*b^2 - 2*c^3*(-2*c + Sqrt[b + a*x]) + b*c*(-9*c + 4*Sqrt[b
+ a*x])) - 8*a*x*(7*b + c*(-6*c + 5*Sqrt[b + a*x]))))/(315*a^3)

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IntegrateAlgebraic [A]  time = 0.09, size = 110, normalized size = 0.99 \begin {gather*} \frac {4 \sqrt {c+\sqrt {b+a x}} \left (315 a^2+315 b^2-336 b c^2+128 c^4+168 b c \sqrt {b+a x}-64 c^3 \sqrt {b+a x}-126 b (b+a x)+48 c^2 (b+a x)-40 c (b+a x)^{3/2}+35 (b+a x)^2\right )}{315 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^2)/(Sqrt[b + a*x]*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

(4*Sqrt[c + Sqrt[b + a*x]]*(315*a^2 + 315*b^2 - 336*b*c^2 + 128*c^4 + 168*b*c*Sqrt[b + a*x] - 64*c^3*Sqrt[b +
a*x] - 126*b*(b + a*x) + 48*c^2*(b + a*x) - 40*c*(b + a*x)^(3/2) + 35*(b + a*x)^2))/(315*a^3)

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fricas [A]  time = 0.70, size = 84, normalized size = 0.76 \begin {gather*} \frac {4 \, {\left (128 \, c^{4} + 35 \, a^{2} x^{2} - 288 \, b c^{2} + 315 \, a^{2} + 224 \, b^{2} + 8 \, {\left (6 \, a c^{2} - 7 \, a b\right )} x - 8 \, {\left (8 \, c^{3} + 5 \, a c x - 16 \, b c\right )} \sqrt {a x + b}\right )} \sqrt {c + \sqrt {a x + b}}}{315 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(a*x+b)^(1/2)/(c+(a*x+b)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

4/315*(128*c^4 + 35*a^2*x^2 - 288*b*c^2 + 315*a^2 + 224*b^2 + 8*(6*a*c^2 - 7*a*b)*x - 8*(8*c^3 + 5*a*c*x - 16*
b*c)*sqrt(a*x + b))*sqrt(c + sqrt(a*x + b))/a^3

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giac [A]  time = 0.18, size = 160, normalized size = 1.44 \begin {gather*} \frac {4 \, {\left (35 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {9}{2}} - 180 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {7}{2}} c + 378 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {c + \sqrt {a x + b}} c^{4} + 315 \, a^{2} \sqrt {c + \sqrt {a x + b}} + 315 \, b^{2} \sqrt {c + \sqrt {a x + b}} - 42 \, {\left (3 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {5}{2}} - 10 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} c + 15 \, \sqrt {c + \sqrt {a x + b}} c^{2}\right )} b\right )}}{315 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(a*x+b)^(1/2)/(c+(a*x+b)^(1/2))^(1/2),x, algorithm="giac")

[Out]

4/315*(35*(c + sqrt(a*x + b))^(9/2) - 180*(c + sqrt(a*x + b))^(7/2)*c + 378*(c + sqrt(a*x + b))^(5/2)*c^2 - 42
0*(c + sqrt(a*x + b))^(3/2)*c^3 + 315*sqrt(c + sqrt(a*x + b))*c^4 + 315*a^2*sqrt(c + sqrt(a*x + b)) + 315*b^2*
sqrt(c + sqrt(a*x + b)) - 42*(3*(c + sqrt(a*x + b))^(5/2) - 10*(c + sqrt(a*x + b))^(3/2)*c + 15*sqrt(c + sqrt(
a*x + b))*c^2)*b)/a^3

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maple [A]  time = 0.08, size = 160, normalized size = 1.44

method	result	size
derivativedivides	\(\frac {\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {9}{2}}}{9}-\frac {16 c \left (c +\sqrt {a x +b}\right )^{\frac {7}{2}}}{7}+\frac {24 c^{2} \left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {8 b \left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {16 c^{3} \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+\frac {16 b c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 c^{4} \sqrt {c +\sqrt {a x +b}}-8 b \,c^{2} \sqrt {c +\sqrt {a x +b}}+4 a^{2} \sqrt {c +\sqrt {a x +b}}+4 b^{2} \sqrt {c +\sqrt {a x +b}}}{a^{3}}\)	\(160\)
default	\(\frac {\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {9}{2}}}{9}-\frac {16 c \left (c +\sqrt {a x +b}\right )^{\frac {7}{2}}}{7}+\frac {24 c^{2} \left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {8 b \left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {16 c^{3} \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+\frac {16 b c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 c^{4} \sqrt {c +\sqrt {a x +b}}-8 b \,c^{2} \sqrt {c +\sqrt {a x +b}}+4 a^{2} \sqrt {c +\sqrt {a x +b}}+4 b^{2} \sqrt {c +\sqrt {a x +b}}}{a^{3}}\)	\(160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(a*x+b)^(1/2)/(c+(a*x+b)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/a^3*(2/9*(c+(a*x+b)^(1/2))^(9/2)-8/7*c*(c+(a*x+b)^(1/2))^(7/2)+12/5*c^2*(c+(a*x+b)^(1/2))^(5/2)-4/5*b*(c+(a*
x+b)^(1/2))^(5/2)-8/3*c^3*(c+(a*x+b)^(1/2))^(3/2)+8/3*b*c*(c+(a*x+b)^(1/2))^(3/2)+2*c^4*(c+(a*x+b)^(1/2))^(1/2
)-4*b*c^2*(c+(a*x+b)^(1/2))^(1/2)+2*a^2*(c+(a*x+b)^(1/2))^(1/2)+2*b^2*(c+(a*x+b)^(1/2))^(1/2))

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maxima [A]  time = 0.31, size = 162, normalized size = 1.46 \begin {gather*} \frac {4 \, {\left (315 \, \sqrt {c + \sqrt {a x + b}} + \frac {35 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {9}{2}} - 180 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {7}{2}} c + 378 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {c + \sqrt {a x + b}} c^{4} + 315 \, b^{2} \sqrt {c + \sqrt {a x + b}} - 42 \, {\left (3 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {5}{2}} - 10 \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} c + 15 \, \sqrt {c + \sqrt {a x + b}} c^{2}\right )} b}{a^{2}}\right )}}{315 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(a*x+b)^(1/2)/(c+(a*x+b)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

4/315*(315*sqrt(c + sqrt(a*x + b)) + (35*(c + sqrt(a*x + b))^(9/2) - 180*(c + sqrt(a*x + b))^(7/2)*c + 378*(c
+ sqrt(a*x + b))^(5/2)*c^2 - 420*(c + sqrt(a*x + b))^(3/2)*c^3 + 315*sqrt(c + sqrt(a*x + b))*c^4 + 315*b^2*sqr
t(c + sqrt(a*x + b)) - 42*(3*(c + sqrt(a*x + b))^(5/2) - 10*(c + sqrt(a*x + b))^(3/2)*c + 15*sqrt(c + sqrt(a*x
+ b))*c^2)*b)/a^2)/a

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+1}{\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/((c + (b + a*x)^(1/2))^(1/2)*(b + a*x)^(1/2)),x)

[Out]

int((x^2 + 1)/((c + (b + a*x)^(1/2))^(1/2)*(b + a*x)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{\sqrt {c + \sqrt {a x + b}} \sqrt {a x + b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)/(a*x+b)**(1/2)/(c+(a*x+b)**(1/2))**(1/2),x)

[Out]

Integral((x**2 + 1)/(sqrt(c + sqrt(a*x + b))*sqrt(a*x + b)), x)

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