Optimal. Leaf size=445 \[ \frac{b \sqrt{c} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{15 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}+\frac{x \sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 \sqrt{c} d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.405257, antiderivative size = 445, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {413, 528, 531, 418, 492, 411} \[ -\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}+\frac{x \sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt{c+d x^2}}+\frac{b \sqrt{c} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \left (103 a^2 b c d^2-15 a^3 d^3-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 \sqrt{c} d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx &=-\frac{(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt{c+d x^2}}+\frac{\int \frac{\left (a+b x^2\right )^{3/2} \left (a b c+b (6 b c-5 a d) x^2\right )}{\sqrt{c+d x^2}} \, dx}{c d}\\ &=-\frac{(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt{c+d x^2}}+\frac{b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 c d^2}+\frac{\int \frac{\sqrt{a+b x^2} \left (-2 a b c (3 b c-5 a d)-b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x^2\right )}{\sqrt{c+d x^2}} \, dx}{5 c d^2}\\ &=-\frac{(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt{c+d x^2}}-\frac{b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 c d^3}+\frac{b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 c d^2}+\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}{15 c d^3}\\ &=-\frac{(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt{c+d x^2}}-\frac{b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 c d^3}+\frac{b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 c d^2}+\frac{\left (a b \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}{15 d^3}+\frac{\left (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 )\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 c 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}}{15 c d^3 \sqrt{c+d x^2}}-\frac{(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt{c+d x^2}}-\frac{b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 c d^3}+\frac{b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 c d^2}+\frac{b \sqrt{c} \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 )}{15 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 (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{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 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}}{15 c d^3 \sqrt{c+d x^2}}-\frac{(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt{c+d x^2}}-\frac{b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 c d^3}+\frac{b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 c d^2}-\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 ) \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 \sqrt{c} 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{b \sqrt{c} \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 )}{15 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 = 1.1849, size = 318, normalized size = 0.71 \[ \frac{4 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (41 a^2 b c d^2-15 a^3 d^3-38 a b^2 c^2 d+12 b^3 c^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (-45 a^2 b c d^2+15 a^3 d^3+a b^2 c d \left (61 c+16 d x^2\right )-3 b^3 c \left (8 c^2+2 c d x^2-d^2 x^4\right )\right )+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (-103 a^2 b c d^2+15 a^3 d^3+128 a b^2 c^2 d-48 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{15 c d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 755, normalized size = 1.7 \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{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{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^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, 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{\left (a + b x^{2}\right )^{\frac{7}{2}}}{\left (c + d x^{2}\right )^{\frac{3}{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{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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