Optimal. Leaf size=253 \[ -\frac {2 B \left (a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {B (2 a-3 i b) (-b+i a)^{5/2} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {B (2 a+3 i b) (b+i a)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d} \]
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Rubi [A] time = 2.46, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3605, 3645, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ -\frac {2 B \left (a^2+3 b^2\right ) \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {B (2 a-3 i b) (-b+i a)^{5/2} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}-\frac {B (2 a+3 i b) (b+i a)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 205
Rule 206
Rule 208
Rule 217
Rule 3605
Rule 3645
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^{5/2} \left (\frac {3 b B}{2 a}+B \tan (c+d x)\right )}{\tan ^{\frac {5}{2}}(c+d x)} \, dx &=-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2}{3} \int \frac {\sqrt {a+b \tan (c+d x)} \left (\frac {3}{2} \left (a^2+3 b^2\right ) B+\frac {3}{4} b \left (a+\frac {3 b^2}{a}\right ) B \tan (c+d x)+\frac {3}{2} b^2 B \tan ^2(c+d x)\right )}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4}{3} \int \frac {\frac {9}{8} b \left (a^2+3 b^2\right ) B-\frac {3 \left (2 a^4+3 a^2 b^2-3 b^4\right ) B \tan (c+d x)}{8 a}+\frac {3}{4} b^3 B \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 \operatorname {Subst}\left (\int \frac {\frac {9}{8} b \left (a^2+3 b^2\right ) B-\frac {3 \left (2 a^4+3 a^2 b^2-3 b^4\right ) B x}{8 a}+\frac {3}{4} b^3 B x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{3 d}\\ &=-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 \operatorname {Subst}\left (\int \left (\frac {3 b^3 B}{4 \sqrt {x} \sqrt {a+b x}}+\frac {3 \left (a b \left (3 a^2+7 b^2\right ) B-\left (2 a^4+3 a^2 b^2-3 b^4\right ) B x\right )}{8 a \sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{3 d}\\ &=-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\operatorname {Subst}\left (\int \frac {a b \left (3 a^2+7 b^2\right ) B-\left (2 a^4+3 a^2 b^2-3 b^4\right ) B x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{2 a d}+\frac {\left (b^3 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\operatorname {Subst}\left (\int \left (\frac {i a b \left (3 a^2+7 b^2\right ) B+\left (2 a^4+3 a^2 b^2-3 b^4\right ) B}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {i a b \left (3 a^2+7 b^2\right ) B-\left (2 a^4+3 a^2 b^2-3 b^4\right ) B}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{2 a d}+\frac {\left (2 b^3 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\left ((a+i b)^3 (2 a-3 i b) B\right ) \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{4 a d}-\frac {\left ((a-i b)^3 (2 a+3 i b) B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{4 a d}+\frac {\left (2 b^3 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\\ &=\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}+\frac {\left ((a+i b)^3 (2 a-3 i b) B\right ) \operatorname {Subst}\left (\int \frac {1}{i-(a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d}-\frac {\left ((a-i b)^3 (2 a+3 i b) B\right ) \operatorname {Subst}\left (\int \frac {1}{i-(-a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d}\\ &=\frac {(i a-b)^{5/2} (2 a-3 i b) B \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d}+\frac {2 b^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(2 a+3 i b) (i a+b)^{5/2} B \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{2 a d}-\frac {2 \left (a^2+3 b^2\right ) B \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}-\frac {b B (a+b \tan (c+d x))^{3/2}}{d \tan ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 4.36, size = 356, normalized size = 1.41 \[ \frac {B \cos (c+d x) (2 a \tan (c+d x)+3 b) \left (4 \sqrt {a} b^{5/2} \tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )-\sqrt {\frac {b \tan (c+d x)}{a}+1} \left (2 a \sqrt {a+b \tan (c+d x)} \left (\left (2 a^2+7 b^2\right ) \tan (c+d x)+a b\right )+\sqrt [4]{-1} (2 a+3 i b) (-a+i b)^{5/2} \tan ^{\frac {3}{2}}(c+d x) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+\sqrt [4]{-1} (a+i b)^{5/2} (2 a-3 i b) \tan ^{\frac {3}{2}}(c+d x) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )\right )}{2 a d \tan ^{\frac {3}{2}}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} (2 a \sin (c+d x)+3 b \cos (c+d x))} \]
Antiderivative was successfully verified.
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fricas [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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giac [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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maple [B] time = 0.89, size = 1490268, normalized size = 5890.39 \[ \text {output too large to display} \]
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 \[ \frac {1}{2} \, \int \frac {{\left (2 \, B \tan \left (d x + c\right ) + \frac {3 \, B b}{a}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\tan \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (B\,\mathrm {tan}\left (c+d\,x\right )+\frac {3\,B\,b}{2\,a}\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {B \left (\int \frac {2 a^{3} \sqrt {a + b \tan {\left (c + d x \right )}}}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \frac {3 b^{3} \sqrt {a + b \tan {\left (c + d x \right )}}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx + \int \frac {6 a b^{2} \sqrt {a + b \tan {\left (c + d x \right )}}}{\tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int 2 a b^{2} \sqrt {a + b \tan {\left (c + d x \right )}} \sqrt {\tan {\left (c + d x \right )}}\, dx + \int \frac {3 a^{2} b \sqrt {a + b \tan {\left (c + d x \right )}}}{\tan ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx + \int \frac {4 a^{2} b \sqrt {a + b \tan {\left (c + d x \right )}}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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