Optimal. Leaf size=291 \[ -\frac {(b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b^2 d^4 n}+\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}+\frac {(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{64 b^{5/2} d^{9/2} n}-\frac {(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n} \]
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Rubi [A] time = 0.32, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {446, 90, 80, 50, 63, 217, 206} \[ \frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}-\frac {(b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b^2 d^4 n}+\frac {(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{64 b^{5/2} d^{9/2} n}-\frac {(3 a d+7 b c) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{3/2}}{\sqrt {c+d x^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 (a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{3/2} \left (-a c-\frac {1}{2} (7 b c+3 a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^n\right )}{4 b d n}\\ &=-\frac {(7 b c+3 a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{48 b^2 d^2 n}\\ &=\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}-\frac {(7 b c+3 a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}-\frac {\left ((b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{64 b^2 d^3 n}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b^2 d^4 n}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}-\frac {(7 b c+3 a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^n\right )}{128 b^2 d^4 n}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b^2 d^4 n}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}-\frac {(7 b c+3 a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^n}\right )}{64 b^3 d^4 n}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b^2 d^4 n}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}-\frac {(7 b c+3 a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^n}}{\sqrt {c+d x^n}}\right )}{64 b^3 d^4 n}\\ &=-\frac {(b c-a d) \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{64 b^2 d^4 n}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{96 b^2 d^3 n}-\frac {(7 b c+3 a d) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{24 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{4 b d n}+\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{64 b^{5/2} d^{9/2} n}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 241, normalized size = 0.83 \[ \frac {3 (b c-a d)^{5/2} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )-b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (9 a^3 d^3+3 a^2 b d^2 \left (5 c-2 d x^n\right )-a b^2 d \left (145 c^2-92 c d x^n+72 d^2 x^{2 n}\right )+b^3 \left (105 c^3-70 c^2 d x^n+56 c d^2 x^{2 n}-48 d^3 x^{3 n}\right )\right )}{192 b^3 d^{9/2} n \sqrt {c+d x^n}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 607, normalized size = 2.09 \[ \left [\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, \sqrt {b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{768 \, b^{3} d^{5} n}, -\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, \sqrt {-b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {-b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{2 \, {\left (b^{2} d^{2} x^{2 \, n} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3 \, n} - 105 \, b^{4} c^{3} d + 145 \, a b^{3} c^{2} d^{2} - 15 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{2 \, n} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 46 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{384 \, b^{3} d^{5} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.93, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{n}+a \right )^{\frac {3}{2}} x^{3 n -1}}{\sqrt {d \,x^{n}+c}}\, dx \]
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 \[ \int \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \]
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 \[ \int \frac {x^{3\,n-1}\,{\left (a+b\,x^n\right )}^{3/2}}{\sqrt {c+d\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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