Optimal. Leaf size=436 \[ -\frac{c^{3/2} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{x \sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right )}{105 b d^3 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d} \]
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Rubi [A] time = 0.489241, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {477, 581, 582, 531, 418, 492, 411} \[ \frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{x \sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right )}{105 b d^3 \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 477
Rule 581
Rule 582
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b x^2\right )^{5/2}}{\sqrt{c+d x^2}} \, dx &=\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d}+\frac{\int \frac{x^2 \sqrt{a+b x^2} \left (-a (3 b c-7 a d)-2 b (3 b c-5 a d) x^2\right )}{\sqrt{c+d x^2}} \, dx}{7 d}\\ &=-\frac{2 b (3 b c-5 a d) x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d}+\frac{\int \frac{x^2 \left (a \left (18 b^2 c^2-45 a b c d+35 a^2 d^2\right )+b \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x^2\right )}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{35 d^2}\\ &=\frac{\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{105 d^3}-\frac{2 b (3 b c-5 a d) x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d}-\frac{\int \frac{a b c \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right )+b \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{105 b d^3}\\ &=\frac{\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{105 d^3}-\frac{2 b (3 b c-5 a d) x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d}-\frac{\left (a c \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{105 d^3}-\frac{\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{105 d^3}\\ &=-\frac{\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt{a+b x^2}}{105 b d^3 \sqrt{c+d x^2}}+\frac{\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{105 d^3}-\frac{2 b (3 b c-5 a d) x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d}-\frac{c^{3/2} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{\left (c \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right )\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{105 b d^3}\\ &=-\frac{\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt{a+b x^2}}{105 b d^3 \sqrt{c+d x^2}}+\frac{\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{105 d^3}-\frac{2 b (3 b c-5 a d) x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d}+\frac{\sqrt{c} \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 b d^{7/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{c^{3/2} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.5944, size = 306, normalized size = 0.7 \[ \frac{4 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (-41 a^2 b c d^2+15 a^3 d^3+38 a b^2 c^2 d-12 b^3 c^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (45 a^2 d^2+a b d \left (45 d x^2-61 c\right )+3 b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (-103 a^2 b c d^2+15 a^3 d^3+128 a b^2 c^2 d-48 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{105 d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 782, normalized size = 1.8 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}} x^{2}}{\sqrt{d x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}\right )} \sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{\sqrt{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}} x^{2}}{\sqrt{d x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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