Optimal. Leaf size=445 \[ \frac{b \sqrt{c} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}+\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 \sqrt{c} d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]
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Rubi [A] time = 1.01467, 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 \[ \frac{b \sqrt{c} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{15 c d^3}+\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{15 c d^3 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 \sqrt{c} d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (6 b c-5 a d)}{5 c d^2}-\frac{x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 143.453, size = 418, normalized size = 0.94 \[ \frac{a^{\frac{3}{2}} \sqrt{b} \sqrt{c + d x^{2}} \left (45 a^{2} d^{2} - 61 a b c d + 24 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 c d^{3} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{b x \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (5 a d - 6 b c\right )}{5 c d^{2}} - \frac{b x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (15 a^{2} d^{2} - 43 a b c d + 24 b^{2} c^{2}\right )}{15 c d^{3}} + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (a d - b c\right )}{c d \sqrt{c + d x^{2}}} - \frac{x \sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right )}{15 c d^{3} \sqrt{c + d x^{2}}} + \frac{\sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{15 \sqrt{c} d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(7/2)/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [C] time = 2.09831, size = 318, normalized size = 0.71 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (15 a^3 d^3-45 a^2 b c d^2+a b^2 c d \left (61 c+16 d x^2\right )-3 b^3 c \left (8 c^2+2 c d x^2-d^2 x^4\right )\right )+4 i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (-15 a^3 d^3+41 a^2 b c d^2-38 a b^2 c^2 d+12 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (15 a^3 d^3-103 a^2 b c d^2+128 a b^2 c^2 d-48 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{15 c d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.061, size = 755, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(7/2)/(d*x^2+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(7/2)/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(7/2)/(d*x^2 + c)^(3/2),x, algorithm="giac")
[Out]