3.249 \(\int \frac {(a+b \sinh ^{-1}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx\)

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

[Out]

-2*(a+b*arcsinh(d*x+c))^3/d/e/(e*(d*x+c))^(1/2)+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 = 0.19, 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 = {} \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(3/2),x]

[Out]

(-2*(a + b*ArcSinh[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)]) + (6*b*Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^2/(S
qrt[e*x]*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}{(c e+d e x)^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e \sqrt {e (c+d x)}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {e x} \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]  time = 19.93, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(3/2),x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(3/2), x]

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fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arsinh}\left (d x + c\right ) + a^{3}\right )} \sqrt {d e x + c e}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, b^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3}}{\sqrt {d x + c} d e^{\frac {3}{2}}} - \frac {2 \, a^{3}}{\sqrt {d e x + c e} d e} + \int \frac {3 \, {\left ({\left (2 \, {\left (c^{3} + c\right )} b^{3} + {\left (a b^{2} d^{3} + 2 \, b^{3} d^{3}\right )} x^{3} + {\left (a c^{3} + a c\right )} b^{2} + 3 \, {\left (a b^{2} c d^{2} + 2 \, b^{3} c d^{2}\right )} x^{2} + {\left (2 \, {\left (3 \, c^{2} d + d\right )} b^{3} + {\left (3 \, a c^{2} d + a d\right )} b^{2}\right )} x + {\left (2 \, b^{3} c^{2} + {\left (a c^{2} + a\right )} b^{2} + {\left (a b^{2} d^{2} + 2 \, b^{3} d^{2}\right )} x^{2} + 2 \, {\left (a b^{2} c d + 2 \, b^{3} c d\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + {\left (a^{2} b d^{3} x^{3} + 3 \, a^{2} b c d^{2} x^{2} + {\left (3 \, a^{2} c^{2} d + a^{2} d\right )} b x + {\left (a^{2} c^{3} + a^{2} c\right )} b + {\left (a^{2} b d^{2} x^{2} + 2 \, a^{2} b c d x + {\left (a^{2} c^{2} + a^{2}\right )} b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )}}{{\left (d^{3} e^{\frac {3}{2}} x^{3} + 3 \, c d^{2} e^{\frac {3}{2}} x^{2} + c^{3} e^{\frac {3}{2}} + c e^{\frac {3}{2}} + {\left (3 \, c^{2} d e^{\frac {3}{2}} + d e^{\frac {3}{2}}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d x + c} + {\left (d^{4} e^{\frac {3}{2}} x^{4} + 4 \, c d^{3} e^{\frac {3}{2}} x^{3} + c^{4} e^{\frac {3}{2}} + c^{2} e^{\frac {3}{2}} + {\left (6 \, c^{2} d^{2} e^{\frac {3}{2}} + d^{2} e^{\frac {3}{2}}\right )} x^{2} + 2 \, {\left (2 \, c^{3} d e^{\frac {3}{2}} + c d e^{\frac {3}{2}}\right )} x\right )} \sqrt {d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^3*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/(sqrt(d*x + c)*d*e^(3/2)) - 2*a^3/(sqrt(d*e*x + c*e)
*d*e) + integrate(3*((2*(c^3 + c)*b^3 + (a*b^2*d^3 + 2*b^3*d^3)*x^3 + (a*c^3 + a*c)*b^2 + 3*(a*b^2*c*d^2 + 2*b
^3*c*d^2)*x^2 + (2*(3*c^2*d + d)*b^3 + (3*a*c^2*d + a*d)*b^2)*x + (2*b^3*c^2 + (a*c^2 + a)*b^2 + (a*b^2*d^2 +
2*b^3*d^2)*x^2 + 2*(a*b^2*c*d + 2*b^3*c*d)*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*x^3 + 3*a^2*b*c*d^2*x^2 + (3*a^2*c^2*d + a^2*d)*b*x + (a^2*c^3 + a^2*c)*b +
(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + (a^2*c^2 + a^2)*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^3*e^(3/2)*x^3 + 3*c*d^2*e^(3/2)*x^2 + c^3*e^(3/2) + c*e^(3/2) + (3*c^2*d*e^(3/
2) + d*e^(3/2))*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (d^4*e^(3/2)*x^4 + 4*c*d^3*e^(3/2)*x^3 +
c^4*e^(3/2) + c^2*e^(3/2) + (6*c^2*d^2*e^(3/2) + d^2*e^(3/2))*x^2 + 2*(2*c^3*d*e^(3/2) + c*d*e^(3/2))*x)*sqrt(
d*x + c)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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