Optimal. Leaf size=311 \[ \frac {2 \left (35 a^2 A-42 a b B-3 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (105 a^3 B+140 a^2 A b-21 a b^2 B+6 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}}-\frac {(-b+i a)^{3/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (7 a B+8 A b) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {(b+i a)^{3/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.48, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3605, 3649, 3616, 3615, 93, 203, 206} \[ \frac {2 \left (35 a^2 A-42 a b B-3 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (140 a^2 A b+105 a^3 B-21 a b^2 B+6 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}}-\frac {(-b+i a)^{3/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (7 a B+8 A b) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {(b+i a)^{3/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 203
Rule 206
Rule 3605
Rule 3615
Rule 3616
Rule 3649
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx &=-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {\frac {1}{2} a (8 A b+7 a B)-\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (6 a A-7 b B) \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {4 \int \frac {\frac {1}{4} a \left (35 a^2 A-3 A b^2-42 a b B\right )+\frac {35}{4} a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+a b (8 A b+7 a B) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{35 a}\\ &=-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {-\frac {1}{8} a \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right )+\frac {105}{8} a^2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac {1}{4} a b \left (35 a^2 A-3 A b^2-42 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^2}\\ &=-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}}-\frac {16 \int \frac {-\frac {105}{16} a^3 \left (a^2 A-A b^2-2 a b B\right )-\frac {105}{16} a^3 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^3}\\ &=-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}}+\frac {1}{2} \left ((a-i b)^2 (A-i B)\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} \left ((a+i b)^2 (A+i B)\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}}+\frac {\left ((a-i b)^2 (A-i B)\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left ((a+i b)^2 (A+i B)\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}}+\frac {\left ((a-i b)^2 (A-i B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}+\frac {\left ((a+i b)^2 (A+i B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\\ &=-\frac {(i a-b)^{3/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(i a+b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (8 A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A-3 A b^2-42 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (140 a^2 A b+6 A b^3+105 a^3 B-21 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \sqrt {\tan (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.40, size = 346, normalized size = 1.11 \[ \frac {-5 a^3 (6 a A-7 b B) \sqrt {a+b \tan (c+d x)}-6 a^3 (7 a B+8 A b) \tan (c+d x) \sqrt {a+b \tan (c+d x)}-35 a^3 b B \sqrt {a+b \tan (c+d x)}+a \tan ^2(c+d x) \left (2 a \left (35 a^2 A-42 a b B-3 A b^2\right ) \sqrt {a+b \tan (c+d x)}-105 (-1)^{3/4} a^2 \tan ^{\frac {3}{2}}(c+d x) \left ((-a+i b)^{3/2} (A-i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+(a+i b)^{3/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )+2 \left (105 a^3 B+140 a^2 A b-21 a b^2 B+6 A b^3\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}\right )}{105 a^3 d \tan ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.04, size = 2400710, normalized size = 7719.32 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\tan \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{{\mathrm {tan}\left (c+d\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________