3.3.57 \(\int \frac {(a+b \sinh ^{-1}(c+d x))^4}{(c e+d e x)^{3/2}} \, dx\) [257]

Optimal. Leaf size=78 \[ -\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e \sqrt {e (c+d x)}}+\frac {8 b \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}},x\right )}{e} \]

[Out]

-2*(a+b*arcsinh(d*x+c))^4/d/e/(e*(d*x+c))^(1/2)+8*b*Unintegrable((a+b*arcsinh(d*x+c))^3/(e*(d*x+c))^(1/2)/(1+(
d*x+c)^2)^(1/2),x)/e

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Rubi [A]
time = 0.12, 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 )^4}{(c e+d e x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2),x]

[Out]

(-2*(a + b*ArcSinh[c + d*x])^4)/(d*e*Sqrt[e*(c + d*x)]) + (8*b*Defer[Subst][Defer[Int][(a + b*ArcSinh[x])^3/(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 )^4}{(c e+d e x)^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^4}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e \sqrt {e (c+d x)}}+\frac {(8 b) \text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {e x} \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]
time = 74.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2),x]

[Out]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsinh \left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(3/2),x)

[Out]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^(3/2),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))^4/(d*e*x+c*e)^(3/2),x, algorithm="maxima")

[Out]

-2*sqrt(d*x + c)*b^4*e^(1/2)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4/(d^2*x*e^2 + c*d*e^2) - 2*a^4*
e^(-1)/(sqrt(d*x*e + c*e)*d) + integrate(2*(2*(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((a*b^3*d^2 + 2*b^4*d^2)*x^2*
e^(1/2) + 2*(a*b^3*c*d + 2*b^4*c*d)*x*e^(1/2) + (2*b^4*c^2 + (c^2 + 1)*a*b^3)*e^(1/2))*sqrt(d*x + c) + ((a*b^3
*d^3 + 2*b^4*d^3)*x^3*e^(1/2) + 3*(a*b^3*c*d^2 + 2*b^4*c*d^2)*x^2*e^(1/2) + ((3*c^2*d + d)*a*b^3 + 2*(3*c^2*d
+ d)*b^4)*x*e^(1/2) + ((c^3 + c)*a*b^3 + 2*(c^3 + c)*b^4)*e^(1/2))*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1))^3 + 3*((a^2*b^2*d^2*x^2*e^(1/2) + 2*a^2*b^2*c*d*x*e^(1/2) + (c^2 + 1)*a^2*b^2*e^(1/2))*sq
rt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*x + c) + (a^2*b^2*d^3*x^3*e^(1/2) + 3*a^2*b^2*c*d^2*x^2*e^(1/2) + (3*c^
2*d + d)*a^2*b^2*x*e^(1/2) + (c^3 + c)*a^2*b^2*e^(1/2))*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x +
c^2 + 1))^2 + 2*((a^3*b*d^2*x^2*e^(1/2) + 2*a^3*b*c*d*x*e^(1/2) + (c^2 + 1)*a^3*b*e^(1/2))*sqrt(d^2*x^2 + 2*c*
d*x + c^2 + 1)*sqrt(d*x + c) + (a^3*b*d^3*x^3*e^(1/2) + 3*a^3*b*c*d^2*x^2*e^(1/2) + (3*c^2*d + d)*a^3*b*x*e^(1
/2) + (c^3 + c)*a^3*b*e^(1/2))*sqrt(d*x + c))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)))/(d^5*x^5*e^2 +
5*c*d^4*x^4*e^2 + (10*c^2*d^3 + d^3)*x^3*e^2 + (10*c^3*d^2 + 3*c*d^2)*x^2*e^2 + (5*c^4*d + 3*c^2*d)*x*e^2 + (
c^5 + c^3)*e^2 + (d^4*x^4*e^2 + 4*c*d^3*x^3*e^2 + (6*c^2*d^2 + d^2)*x^2*e^2 + 2*(2*c^3*d + c*d)*x*e^2 + (c^4 +
c^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), 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))^4/(d*e*x+c*e)^(3/2),x, algorithm="fricas")

[Out]

integral((b^4*arcsinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2*arcsinh(d*x + c)^2 + 4*a^3*b*arcsinh
(d*x + c) + a^4)*sqrt(d*x + c)*e^(-3/2)/(d^2*x^2 + 2*c*d*x + c^2), 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 )^{4}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(d*x+c))**4/(d*e*x+c*e)**(3/2),x)

[Out]

Integral((a + b*asinh(c + d*x))**4/(e*(c + d*x))**(3/2), 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))^4/(d*e*x+c*e)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^4/(d*e*x + c*e)^(3/2), 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 )}^4}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^(3/2),x)

[Out]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^(3/2), x)

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