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

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

[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 = 0.18, 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 = {} \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {c e+d e x}} \, dx \]

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 {\operatorname {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) \operatorname {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 = 9.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {c e+d e x}} \, dx \]

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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fricas [A]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arsinh}\left (d x + c\right ) + a^{3}}{\sqrt {d e x + c e}}, x\right ) \]

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)/sqrt(d*e*x + c
*e), x)

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

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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maple [F(-2)]  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 \[ \frac {2 \, \sqrt {d e x + c e} a^{3}}{d e} + \frac {2 \, {\left (b^{3} d \sqrt {e} x + b^{3} c \sqrt {e}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3}}{\sqrt {d x + c} d e} - \int \frac {3 \, {\left ({\left (2 \, {\left (c^{3} \sqrt {e} + c \sqrt {e}\right )} b^{3} - {\left (a b^{2} d^{3} \sqrt {e} - 2 \, b^{3} d^{3} \sqrt {e}\right )} x^{3} - {\left (a c^{3} \sqrt {e} + a c \sqrt {e}\right )} b^{2} - 3 \, {\left (a b^{2} c d^{2} \sqrt {e} - 2 \, b^{3} c d^{2} \sqrt {e}\right )} x^{2} + {\left (2 \, {\left (3 \, c^{2} d \sqrt {e} + d \sqrt {e}\right )} b^{3} - {\left (3 \, a c^{2} d \sqrt {e} + a d \sqrt {e}\right )} b^{2}\right )} x + {\left (2 \, b^{3} c^{2} \sqrt {e} - {\left (a c^{2} \sqrt {e} + a \sqrt {e}\right )} b^{2} - {\left (a b^{2} d^{2} \sqrt {e} - 2 \, b^{3} d^{2} \sqrt {e}\right )} x^{2} - 2 \, {\left (a b^{2} c d \sqrt {e} - 2 \, b^{3} c d \sqrt {e}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} - {\left (a^{2} b d^{3} \sqrt {e} x^{3} + 3 \, a^{2} b c d^{2} \sqrt {e} x^{2} + {\left (3 \, a^{2} c^{2} d \sqrt {e} + a^{2} d \sqrt {e}\right )} b x + {\left (a^{2} c^{3} \sqrt {e} + a^{2} c \sqrt {e}\right )} b + {\left (a^{2} b d^{2} \sqrt {e} x^{2} + 2 \, a^{2} b c d \sqrt {e} x + {\left (a^{2} c^{2} \sqrt {e} + a^{2} \sqrt {e}\right )} b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )}}{{\left (d^{2} e x^{2} + 2 \, c d e x + c^{2} e + e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} + {\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + c^{3} e + c e + {\left (3 \, c^{2} d e + d e\right )} x\right )} \sqrt {d x + c}}\,{d x} \]

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

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

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

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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