Optimal. Leaf size=325 \[ -\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} d \left (a^2+b^2\right )}+\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )}+\frac{(b (A-B)-a (A+B)) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}-\frac{(b (A-B)-a (A+B)) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}+\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d} \]
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Rubi [A] time = 0.979272, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {3607, 3647, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} d \left (a^2+b^2\right )}+\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )}+\frac{(b (A-B)-a (A+B)) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}-\frac{(b (A-B)-a (A+B)) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )}+\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 3607
Rule 3647
Rule 3653
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{2 \int \frac{\sqrt{\tan (c+d x)} \left (-\frac{3 a B}{2}-\frac{3}{2} b B \tan (c+d x)+\frac{3}{2} (A b-a B) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 b}\\ &=\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{4 \int \frac{-\frac{3}{4} a (A b-a B)-\frac{3}{4} A b^2 \tan (c+d x)-\frac{3}{4} \left (a A b-a^2 B+b^2 B\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 b^2}\\ &=\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{4 \int \frac{-\frac{3}{4} b^2 (A b-a B)-\frac{3}{4} b^2 (a A+b B) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )}-\frac{\left (a^3 (A b-a B)\right ) \int \frac{1+\tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{8 \operatorname{Subst}\left (\int \frac{-\frac{3}{4} b^2 (A b-a B)-\frac{3}{4} b^2 (a A+b B) x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{3 b^2 \left (a^2+b^2\right ) d}-\frac{\left (a^3 (A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}-\frac{\left (2 a^3 (A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac{(b (A-B)-a (A+B)) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac{(a (A-B)+b (A+B)) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}+\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{(b (A-B)-a (A+B)) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}+\frac{(b (A-B)-a (A+B)) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}-\frac{(a (A-B)+b (A+B)) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac{(a (A-B)+b (A+B)) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}+\frac{(b (A-B)-a (A+B)) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}-\frac{(b (A-B)-a (A+B)) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}+\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}-\frac{(a (A-B)+b (A+B)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right ) d}+\frac{(a (A-B)+b (A+B)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right ) d}\\ &=\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right ) d}-\frac{(a (A-B)+b (A+B)) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right ) d}-\frac{2 a^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} \left (a^2+b^2\right ) d}+\frac{(b (A-B)-a (A+B)) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}-\frac{(b (A-B)-a (A+B)) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right ) d}+\frac{2 (A b-a B) \sqrt{\tan (c+d x)}}{b^2 d}+\frac{2 B \tan ^{\frac{3}{2}}(c+d x)}{3 b d}\\ \end{align*}
Mathematica [C] time = 1.11041, size = 187, normalized size = 0.58 \[ \frac{2 \sqrt{b} \left (a^2+b^2\right ) \sqrt{\tan (c+d x)} (-3 a B+3 A b+b B \tan (c+d x))+6 a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )-3 \sqrt [4]{-1} b^{5/2} (a+i b) (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )+3 \sqrt [4]{-1} b^{5/2} (b+i a) (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )}{3 b^{5/2} d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 666, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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