3.3.48 \(\int \frac {(a+b \sinh ^{-1}(c+d x))^3}{\sqrt {c e+d e x}} \, dx\) [248]

Optimal. Leaf size=78 \[ \frac {2 \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e}-\frac {6 b \text {Int}\left (\frac {\sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt {1+(c+d x)^2}},x\right )}{e} \]

[Out]

2*(a+b*arcsinh(d*x+c))^3*(e*(d*x+c))^(1/2)/d/e-6*b*Unintegrable((a+b*arcsinh(d*x+c))^2*(e*(d*x+c))^(1/2)/(1+(d
*x+c)^2)^(1/2),x)/e

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Rubi [A]
time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {c e+d e x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x],x]

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcSinh[c + d*x])^3)/(d*e) - (6*b*Defer[Subst][Defer[Int][(Sqrt[e*x]*(a + b*ArcSin
h[x])^2)/Sqrt[1 + x^2], x], x, c + d*x])/(d*e)

Rubi steps

\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {c e+d e x}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \sqrt {e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e}-\frac {(6 b) \text {Subst}\left (\int \frac {\sqrt {e x} \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]
time = 131.87, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {c e+d e x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x],x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x], x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{3}}{\sqrt {d e x +c e}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(d*x*e + c*e)*a^3*e^(-1)/d + 2*(b^3*d*x*e^(1/2) + b^3*c*e^(1/2))*e^(-1)*log(d*x + c + sqrt(d^2*x^2 + 2*c
*d*x + c^2 + 1))^3/(sqrt(d*x + c)*d) + integrate(3*(((a*b^2*d^3 - 2*b^3*d^3)*x^3*e^(1/2) + 3*(a*b^2*c*d^2 - 2*
b^3*c*d^2)*x^2*e^(1/2) + ((3*c^2*d + d)*a*b^2 - 2*(3*c^2*d + d)*b^3)*x*e^(1/2) + ((c^3 + c)*a*b^2 - 2*(c^3 + c
)*b^3)*e^(1/2) + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((a*b^2*d^2 - 2*b^3*d^2)*x^2*e^(1/2) + 2*(a*b^2*c*d - 2*b^3
*c*d)*x*e^(1/2) - (2*b^3*c^2 - (c^2 + 1)*a*b^2)*e^(1/2)))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 +
(a^2*b*d^3*x^3*e^(1/2) + 3*a^2*b*c*d^2*x^2*e^(1/2) + (3*c^2*d + d)*a^2*b*x*e^(1/2) + (c^3 + c)*a^2*b*e^(1/2)
+ (a^2*b*d^2*x^2*e^(1/2) + 2*a^2*b*c*d*x*e^(1/2) + (c^2 + 1)*a^2*b*e^(1/2))*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))
*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)))/((d^2*x^2*e + 2*c*d*x*e + (c^2 + 1)*e)*sqrt(d^2*x^2 + 2*c*d
*x + c^2 + 1)*sqrt(d*x + c) + (d^3*x^3*e + 3*c*d^2*x^2*e + (3*c^2*d + d)*x*e + (c^3 + c)*e)*sqrt(d*x + c)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

integral((b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*arcsinh(d*x + c) + a^3)*e^(-1/2)/sqrt(
d*x + c), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}}{\sqrt {e \left (c + d x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(d*x+c))**3/(d*e*x+c*e)**(1/2),x)

[Out]

Integral((a + b*asinh(c + d*x))**3/sqrt(e*(c + d*x)), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(d*x+c))^3/(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^3/sqrt(d*e*x + c*e), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{\sqrt {c\,e+d\,e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c + d*x))^3/(c*e + d*e*x)^(1/2),x)

[Out]

int((a + b*asinh(c + d*x))^3/(c*e + d*e*x)^(1/2), x)

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