Optimal. Leaf size=290 \[ -\frac {304 b^2 d x}{3675 c^4}-\frac {152 b^2 d x^3}{11025 c^2}-\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7+\frac {32 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{525 c^5}+\frac {16 b d x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{525 c^3}+\frac {4 b d x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{175 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{21 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{49 c^5}+\frac {2}{35} d x^5 (a+b \text {ArcSin}(c x))^2+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2 \]
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Rubi [A]
time = 0.31, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4787, 4723,
4795, 4767, 8, 30, 272, 45, 4779, 12} \begin {gather*} \frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {4 b d x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{175 c}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{49 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{21 c^5}+\frac {32 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{525 c^5}+\frac {16 b d x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{525 c^3}+\frac {2}{35} d x^5 (a+b \text {ArcSin}(c x))^2-\frac {304 b^2 d x}{3675 c^4}+\frac {2}{343} b^2 c^2 d x^7-\frac {152 b^2 d x^3}{11025 c^2}-\frac {38 b^2 d x^5}{6125} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 45
Rule 272
Rule 4723
Rule 4767
Rule 4779
Rule 4787
Rule 4795
Rubi steps
\begin {align*} \int x^4 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} (2 d) \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{7} (2 b c d) \int x^5 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{35} (4 b c d) \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{7} \left (2 b^2 c^2 d\right ) \int \frac {-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx\\ &=\frac {4 b d x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{175} \left (4 b^2 d\right ) \int x^4 \, dx+\frac {\left (2 b^2 d\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+15 c^6 x^6\right ) \, dx}{735 c^4}-\frac {(16 b d) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{175 c}\\ &=-\frac {16 b^2 d x}{735 c^4}-\frac {8 b^2 d x^3}{2205 c^2}-\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7+\frac {16 b d x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac {4 b d x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {(32 b d) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{525 c^3}-\frac {\left (16 b^2 d\right ) \int x^2 \, dx}{525 c^2}\\ &=-\frac {16 b^2 d x}{735 c^4}-\frac {152 b^2 d x^3}{11025 c^2}-\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7+\frac {32 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^5}+\frac {16 b d x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac {4 b d x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (32 b^2 d\right ) \int 1 \, dx}{525 c^4}\\ &=-\frac {304 b^2 d x}{3675 c^4}-\frac {152 b^2 d x^3}{11025 c^2}-\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7+\frac {32 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^5}+\frac {16 b d x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{525 c^3}+\frac {4 b d x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{175 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{21 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{49 c^5}+\frac {2}{35} d x^5 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 203, normalized size = 0.70 \begin {gather*} -\frac {d \left (11025 a^2 c^5 x^5 \left (-7+5 c^2 x^2\right )+210 a b \sqrt {1-c^2 x^2} \left (-152-76 c^2 x^2-57 c^4 x^4+75 c^6 x^6\right )+b^2 \left (31920 c x+5320 c^3 x^3+2394 c^5 x^5-2250 c^7 x^7\right )+210 b \left (105 a c^5 x^5 \left (-7+5 c^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (-152-76 c^2 x^2-57 c^4 x^4+75 c^6 x^6\right )\right ) \text {ArcSin}(c x)+11025 b^2 c^5 x^5 \left (-7+5 c^2 x^2\right ) \text {ArcSin}(c x)^2\right )}{385875 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 276, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {-d \,a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 \arcsin \left (c x \right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right ) \sqrt {-c^{2} x^{2}+1}}{75}+\frac {38 c^{5} x^{5}}{6125}+\frac {152 c^{3} x^{3}}{11025}+\frac {304 c x}{3675}+\frac {\arcsin \left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {2 \arcsin \left (c x \right ) \left (5 c^{6} x^{6}+6 c^{4} x^{4}+8 c^{2} x^{2}+16\right ) \sqrt {-c^{2} x^{2}+1}}{245}-\frac {2 c^{7} x^{7}}{343}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(276\) |
default | \(\frac {-d \,a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 \arcsin \left (c x \right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right ) \sqrt {-c^{2} x^{2}+1}}{75}+\frac {38 c^{5} x^{5}}{6125}+\frac {152 c^{3} x^{3}}{11025}+\frac {304 c x}{3675}+\frac {\arcsin \left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {2 \arcsin \left (c x \right ) \left (5 c^{6} x^{6}+6 c^{4} x^{4}+8 c^{2} x^{2}+16\right ) \sqrt {-c^{2} x^{2}+1}}{245}-\frac {2 c^{7} x^{7}}{343}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(276\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 453, normalized size = 1.56 \begin {gather*} -\frac {1}{7} \, b^{2} c^{2} d x^{7} \arcsin \left (c x\right )^{2} - \frac {1}{7} \, a^{2} c^{2} d x^{7} + \frac {1}{5} \, b^{2} d x^{5} \arcsin \left (c x\right )^{2} + \frac {1}{5} \, a^{2} d x^{5} - \frac {2}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{2} d - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{2} d + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d + \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.68, size = 229, normalized size = 0.79 \begin {gather*} -\frac {1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d x^{7} - 63 \, {\left (1225 \, a^{2} - 38 \, b^{2}\right )} c^{5} d x^{5} + 5320 \, b^{2} c^{3} d x^{3} + 31920 \, b^{2} c d x + 11025 \, {\left (5 \, b^{2} c^{7} d x^{7} - 7 \, b^{2} c^{5} d x^{5}\right )} \arcsin \left (c x\right )^{2} + 22050 \, {\left (5 \, a b c^{7} d x^{7} - 7 \, a b c^{5} d x^{5}\right )} \arcsin \left (c x\right ) + 210 \, {\left (75 \, a b c^{6} d x^{6} - 57 \, a b c^{4} d x^{4} - 76 \, a b c^{2} d x^{2} - 152 \, a b d + {\left (75 \, b^{2} c^{6} d x^{6} - 57 \, b^{2} c^{4} d x^{4} - 76 \, b^{2} c^{2} d x^{2} - 152 \, b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{385875 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.20, size = 388, normalized size = 1.34 \begin {gather*} \begin {cases} - \frac {a^{2} c^{2} d x^{7}}{7} + \frac {a^{2} d x^{5}}{5} - \frac {2 a b c^{2} d x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {2 a b c d x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {2 a b d x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {38 a b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225 c} + \frac {152 a b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675 c^{3}} + \frac {304 a b d \sqrt {- c^{2} x^{2} + 1}}{3675 c^{5}} - \frac {b^{2} c^{2} d x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{2} d x^{7}}{343} - \frac {2 b^{2} c d x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{49} + \frac {b^{2} d x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {38 b^{2} d x^{5}}{6125} + \frac {38 b^{2} d x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{1225 c} - \frac {152 b^{2} d x^{3}}{11025 c^{2}} + \frac {152 b^{2} d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3675 c^{3}} - \frac {304 b^{2} d x}{3675 c^{4}} + \frac {304 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{5}}{5} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 495, normalized size = 1.71 \begin {gather*} -\frac {1}{7} \, a^{2} c^{2} d x^{7} + \frac {1}{5} \, a^{2} d x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d x \arcsin \left (c x\right )^{2}}{7 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d x \arcsin \left (c x\right )}{7 \, c^{4}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d x}{343 \, c^{4}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{49 \, c^{5}} + \frac {484 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x}{42875 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right )}{35 \, c^{4}} + \frac {2 \, b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d}{49 \, c^{5}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{175 \, c^{5}} - \frac {3358 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x}{385875 \, c^{4}} + \frac {4 \, a b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d}{175 \, c^{5}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d \arcsin \left (c x\right )}{105 \, c^{5}} - \frac {37384 \, b^{2} d x}{385875 \, c^{4}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d}{105 \, c^{5}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{35 \, c^{5}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d}{35 \, c^{5}} \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 = -0.00 \begin {gather*} \int x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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