Optimal. Leaf size=198 \[ \frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{2 d}+\frac {3 b^2 e (a+b \text {ArcSin}(c+d x))^2}{4 d}-\frac {3 b^2 e (c+d x)^2 (a+b \text {ArcSin}(c+d x))^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^3}{d}-\frac {e (a+b \text {ArcSin}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {ArcSin}(c+d x))^4}{2 d} \]
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Rubi [A]
time = 0.22, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4889, 12, 4723,
4795, 4737, 30} \begin {gather*} -\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{2 d}-\frac {3 b^2 e (c+d x)^2 (a+b \text {ArcSin}(c+d x))^2}{2 d}+\frac {3 b^2 e (a+b \text {ArcSin}(c+d x))^2}{4 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^3}{d}+\frac {e (c+d x)^2 (a+b \text {ArcSin}(c+d x))^4}{2 d}-\frac {e (a+b \text {ArcSin}(c+d x))^4}{4 d}+\frac {3 b^4 e (c+d x)^2}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 4723
Rule 4737
Rule 4795
Rule 4889
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{2 d}-\frac {(2 b e) \text {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^3}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{2 d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}-\frac {3 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{2 d}+\frac {\left (3 b^3 e\right ) \text {Subst}\left (\int \frac {a+b \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (3 b^4 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{2 d}+\frac {3 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^2}{4 d}-\frac {3 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{2 d}+\frac {b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^3}{d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^4}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^4}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 163, normalized size = 0.82 \begin {gather*} -\frac {e \left (-4 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^3+(a+b \text {ArcSin}(c+d x))^4-2 (c+d x)^2 (a+b \text {ArcSin}(c+d x))^4+3 b^2 \left (-b^2 (c+d x)^2+2 b (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))-(a+b \text {ArcSin}(c+d x))^2+2 (c+d x)^2 (a+b \text {ArcSin}(c+d x))^2\right )\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(411\) vs.
\(2(182)=364\).
time = 0.06, size = 412, normalized size = 2.08
method | result | size |
derivativedivides | \(\frac {\frac {e \left (d x +c \right )^{2} a^{4}}{2}+e \,b^{4} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{4}}{2}+\arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}-\frac {3 \arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}+\frac {3 \arcsin \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}-\frac {3 \arcsin \left (d x +c \right )^{4}}{4}\right )+4 e a \,b^{3} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{3}}{2}+\frac {3 \arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{4}-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsin \left (d x +c \right )}{8}-\frac {\arcsin \left (d x +c \right )^{3}}{2}\right )+6 e \,a^{2} b^{2} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\) | \(412\) |
default | \(\frac {\frac {e \left (d x +c \right )^{2} a^{4}}{2}+e \,b^{4} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{4}}{2}+\arcsin \left (d x +c \right )^{3} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}-\frac {3 \arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}+\frac {3 \arcsin \left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}-\frac {3 \arcsin \left (d x +c \right )^{4}}{4}\right )+4 e a \,b^{3} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{3}}{2}+\frac {3 \arcsin \left (d x +c \right )^{2} \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{4}-\frac {3 \left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsin \left (d x +c \right )}{8}-\frac {\arcsin \left (d x +c \right )^{3}}{2}\right )+6 e \,a^{2} b^{2} \left (\frac {\left (\left (d x +c \right )^{2}-1\right ) \arcsin \left (d x +c \right )^{2}}{2}+\frac {\arcsin \left (d x +c \right ) \left (\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}+\arcsin \left (d x +c \right )\right )}{2}-\frac {\arcsin \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{2}}{4}\right )+4 e \,a^{3} b \left (\frac {\left (d x +c \right )^{2} \arcsin \left (d x +c \right )}{2}+\frac {\left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{4}-\frac {\arcsin \left (d x +c \right )}{4}\right )}{d}\) | \(412\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 471 vs.
\(2 (189) = 378\).
time = 2.15, size = 471, normalized size = 2.38 \begin {gather*} \frac {{\left (2 \, b^{4} d^{2} x^{2} + 4 \, b^{4} c d x + 2 \, b^{4} c^{2} - b^{4}\right )} \arcsin \left (d x + c\right )^{4} e + 4 \, {\left (2 \, a b^{3} d^{2} x^{2} + 4 \, a b^{3} c d x + 2 \, a b^{3} c^{2} - a b^{3}\right )} \arcsin \left (d x + c\right )^{3} e + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} d^{2} x^{2} - 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} c d x + 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} c^{2}\right )} \arcsin \left (d x + c\right )^{2} e + 2 \, {\left (2 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} d^{2} x^{2} - 2 \, a^{3} b + 3 \, a b^{3} + 4 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c d x + 2 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c^{2}\right )} \arcsin \left (d x + c\right ) e + {\left ({\left (2 \, a^{4} - 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} x^{2} + 2 \, {\left (2 \, a^{4} - 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d x\right )} e + 2 \, {\left (2 \, {\left (b^{4} d x + b^{4} c\right )} \arcsin \left (d x + c\right )^{3} e + 6 \, {\left (a b^{3} d x + a b^{3} c\right )} \arcsin \left (d x + c\right )^{2} e + 3 \, {\left ({\left (2 \, a^{2} b^{2} - b^{4}\right )} d x + {\left (2 \, a^{2} b^{2} - b^{4}\right )} c\right )} \arcsin \left (d x + c\right ) e + {\left ({\left (2 \, a^{3} b - 3 \, a b^{3}\right )} d x + {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} c\right )} e\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1027 vs.
\(2 (178) = 356\).
time = 0.60, size = 1027, normalized size = 5.19 \begin {gather*} \begin {cases} a^{4} c e x + \frac {a^{4} d e x^{2}}{2} + \frac {2 a^{3} b c^{2} e \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a^{3} b c e x \operatorname {asin}{\left (c + d x \right )} + \frac {a^{3} b c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + 2 a^{3} b d e x^{2} \operatorname {asin}{\left (c + d x \right )} + a^{3} b e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} - \frac {a^{3} b e \operatorname {asin}{\left (c + d x \right )}}{d} + \frac {3 a^{2} b^{2} c^{2} e \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} c e x \operatorname {asin}^{2}{\left (c + d x \right )} - 3 a^{2} b^{2} c e x + \frac {3 a^{2} b^{2} c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} + 3 a^{2} b^{2} d e x^{2} \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {3 a^{2} b^{2} d e x^{2}}{2} + 3 a^{2} b^{2} e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )} - \frac {3 a^{2} b^{2} e \operatorname {asin}^{2}{\left (c + d x \right )}}{2 d} + \frac {2 a b^{3} c^{2} e \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {3 a b^{3} c^{2} e \operatorname {asin}{\left (c + d x \right )}}{d} + 4 a b^{3} c e x \operatorname {asin}^{3}{\left (c + d x \right )} - 6 a b^{3} c e x \operatorname {asin}{\left (c + d x \right )} + \frac {3 a b^{3} c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )}}{d} - \frac {3 a b^{3} c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{2 d} + 2 a b^{3} d e x^{2} \operatorname {asin}^{3}{\left (c + d x \right )} - 3 a b^{3} d e x^{2} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{3} e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (c + d x \right )} - \frac {3 a b^{3} e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{2} - \frac {a b^{3} e \operatorname {asin}^{3}{\left (c + d x \right )}}{d} + \frac {3 a b^{3} e \operatorname {asin}{\left (c + d x \right )}}{2 d} + \frac {b^{4} c^{2} e \operatorname {asin}^{4}{\left (c + d x \right )}}{2 d} - \frac {3 b^{4} c^{2} e \operatorname {asin}^{2}{\left (c + d x \right )}}{2 d} + b^{4} c e x \operatorname {asin}^{4}{\left (c + d x \right )} - 3 b^{4} c e x \operatorname {asin}^{2}{\left (c + d x \right )} + \frac {3 b^{4} c e x}{2} + \frac {b^{4} c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c + d x \right )}}{d} - \frac {3 b^{4} c e \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{2 d} + \frac {b^{4} d e x^{2} \operatorname {asin}^{4}{\left (c + d x \right )}}{2} - \frac {3 b^{4} d e x^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{2} + \frac {3 b^{4} d e x^{2}}{4} + b^{4} e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (c + d x \right )} - \frac {3 b^{4} e x \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{2} - \frac {b^{4} e \operatorname {asin}^{4}{\left (c + d x \right )}}{4 d} + \frac {3 b^{4} e \operatorname {asin}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 533 vs.
\(2 (182) = 364\).
time = 0.47, size = 533, normalized size = 2.69 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e \arcsin \left (d x + c\right )^{4}}{2 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{4} e \arcsin \left (d x + c\right )^{3}}{d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} e \arcsin \left (d x + c\right )^{3}}{d} + \frac {b^{4} e \arcsin \left (d x + c\right )^{4}}{4 \, d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{3} e \arcsin \left (d x + c\right )^{2}}{d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} e \arcsin \left (d x + c\right )^{2}}{d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e \arcsin \left (d x + c\right )^{2}}{2 \, d} + \frac {a b^{3} e \arcsin \left (d x + c\right )^{3}}{d} + \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{2} b^{2} e \arcsin \left (d x + c\right )}{d} - \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{4} e \arcsin \left (d x + c\right )}{2 \, d} + \frac {2 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{3} b e \arcsin \left (d x + c\right )}{d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a b^{3} e \arcsin \left (d x + c\right )}{d} + \frac {3 \, a^{2} b^{2} e \arcsin \left (d x + c\right )^{2}}{2 \, d} - \frac {3 \, b^{4} e \arcsin \left (d x + c\right )^{2}}{4 \, d} + \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a^{3} b e}{d} - \frac {3 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} a b^{3} e}{2 \, d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a^{4} e}{2 \, d} - \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} a^{2} b^{2} e}{2 \, d} + \frac {3 \, {\left ({\left (d x + c\right )}^{2} - 1\right )} b^{4} e}{4 \, d} + \frac {a^{3} b e \arcsin \left (d x + c\right )}{d} - \frac {3 \, a b^{3} e \arcsin \left (d x + c\right )}{2 \, d} - \frac {3 \, a^{2} b^{2} e}{4 \, d} + \frac {3 \, b^{4} e}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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