Optimal. Leaf size=358 \[ \frac {(b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{128 b^2 d^5 n}-\frac {(b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n}-\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n} \]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \begin {gather*} \frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {(b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {(b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{128 b^2 d^5 n}-\frac {(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{128 b^{5/2} d^{11/2} n}-\frac {3 (a d+3 b c) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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 )^{5/2}}{\sqrt {c+d x^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{n}\\ &=\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2} \left (-a c-\frac {3}{2} (3 b c+a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^n\right )}{5 b d n}\\ &=-\frac {3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{80 b^2 d^2 n}\\ &=\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}-\frac {\left ((b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^n\right )}{96 b^2 d^3 n}\\ &=-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}+\frac {\left ((b c-a d)^2 \left (63 b^2 c^2+14 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 )}{128 b^2 d^4 n}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{128 b^2 d^5 n}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 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 )}{256 b^2 d^5 n}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{128 b^2 d^5 n}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 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 )}{128 b^3 d^5 n}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{128 b^2 d^5 n}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 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 )}{128 b^3 d^5 n}\\ &=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^n} \sqrt {c+d x^n}}{128 b^2 d^5 n}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{192 b^2 d^4 n}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^n\right )^{5/2} \sqrt {c+d x^n}}{240 b^2 d^3 n}-\frac {3 (3 b c+a d) \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{40 b^2 d^2 n}+\frac {x^n \left (a+b x^n\right )^{7/2} \sqrt {c+d x^n}}{5 b d n}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 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 )}{128 b^{5/2} d^{11/2} n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.02, size = 274, normalized size = 0.77 \begin {gather*} \frac {\sqrt {c+d x^n} \left (\frac {5 (b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (-\frac {16 d^3 \left (a+b x^n\right )^3}{15 (a d-b c)^3}-\frac {4 d^2 \left (a+b x^n\right )^2}{3 (b c-a d)^2}-\frac {2 d \left (a+b x^n\right )}{a d-b c}-\frac {2 \sqrt {d} \sqrt {a+b x^n} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}}}\right )}{4 b d^5}-\frac {24 (a d+3 b c) \left (a+b x^n\right )^4}{b d}+64 x^n \left (a+b x^n\right )^4\right )}{320 b d n \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{-1+3 n} \left (a+b x^n\right )^{5/2}}{\sqrt {c+d x^n}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 771, normalized size = 2.15 \begin {gather*} \left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{4 \, n} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3 \, n} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2 \, n} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{7680 \, b^{3} d^{6} n}, \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{4 \, n} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3 \, n} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2 \, n} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{n}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{3840 \, b^{3} d^{6} n}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.96, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{n}+a \right )^{\frac {5}{2}} x^{3 n -1}}{\sqrt {d \,x^{n}+c}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{n} + a\right )}^{\frac {5}{2}} x^{3 \, n - 1}}{\sqrt {d x^{n} + c}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{3\,n-1}\,{\left (a+b\,x^n\right )}^{5/2}}{\sqrt {c+d\,x^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________