3.305 \(\int \frac {(a+b \sin ^{-1}(c+d x))^4}{\sqrt {c e+d e x}} \, dx\)

Optimal. Leaf size=80 \[ \frac {2 \sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d e}-\frac {8 b \text {Int}\left (\frac {\sqrt {e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^3}{\sqrt {1-(c+d x)^2}},x\right )}{e} \]

[Out]

2*(a+b*arcsin(d*x+c))^4*(e*(d*x+c))^(1/2)/d/e-8*b*Unintegrable((a+b*arcsin(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 = 0.18, 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 = {} \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^4}{\sqrt {c e+d e x}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x],x]

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcSin[c + d*x])^4)/(d*e) - (8*b*Defer[Subst][Defer[Int][(Sqrt[e*x]*(a + b*ArcSin[
x])^3)/Sqrt[1 - x^2], x], x, c + d*x])/(d*e)

Rubi steps

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

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Mathematica [A]  time = 10.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sin ^{-1}(c+d x)\right )^4}{\sqrt {c e+d e x}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x],x]

[Out]

Integrate[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x], x]

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}}{\sqrt {d e x + c e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

integral((b^4*arcsin(d*x + c)^4 + 4*a*b^3*arcsin(d*x + c)^3 + 6*a^2*b^2*arcsin(d*x + c)^2 + 4*a^3*b*arcsin(d*x
+ c) + a^4)/sqrt(d*e*x + c*e), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{\sqrt {d e x + c e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)^4/sqrt(d*e*x + c*e), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsin \left (d x +c \right )\right )^{4}}{\sqrt {d e x +c e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x+c))^4/(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4}{\sqrt {c\,e+d\,e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(1/2),x)

[Out]

int((a + b*asin(c + d*x))^4/(c*e + d*e*x)^(1/2), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x+c))**4/(d*e*x+c*e)**(1/2),x)

[Out]

Exception raised: TypeError

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