Optimal. Leaf size=436 \[ \frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{c^{3/2} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{105 b d^3 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d} \]
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Rubi [A] time = 1.27749, 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 \[ \frac{x \sqrt{a+b x^2} \sqrt{c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{105 d^3}-\frac{c^{3/2} \sqrt{a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right )}{105 b d^3 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{105 b d^{7/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 b x^3 \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-5 a d)}{35 d^2}+\frac{b x^3 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{7 d} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]
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Rubi in Sympy [A] time = 149.728, size = 410, normalized size = 0.94 \[ \frac{b x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{7 d} + \frac{2 b x^{3} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (5 a d - 3 b c\right )}{35 d^{2}} - \frac{c^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (45 a^{2} d^{2} - 61 a b c d + 24 b^{2} c^{2}\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{105 d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (45 a^{2} d^{2} - 61 a b c d + 24 b^{2} c^{2}\right )}{105 d^{3}} - \frac{\sqrt{c} \sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{105 b d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{a + b x^{2}} \left (15 a^{3} d^{3} - 103 a^{2} b c d^{2} + 128 a b^{2} c^{2} d - 48 b^{3} c^{3}\right )}{105 b d^{3} \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
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Mathematica [C] time = 1.10101, size = 306, normalized size = 0.7 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (45 a^2 d^2+a b d \left (45 d x^2-61 c\right )+3 b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )+4 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (15 a^3 d^3-41 a^2 b c d^2+38 a b^2 c^2 d-12 b^3 c^3\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (15 a^3 d^3-103 a^2 b c d^2+128 a b^2 c^2 d-48 b^3 c^3\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{105 d^4 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]
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Maple [A] time = 0.03, size = 782, normalized size = 1.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}} x^{2}}{\sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}\right )} \sqrt{b x^{2} + a}}{\sqrt{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{\sqrt{c + d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}} x^{2}}{\sqrt{d x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)*x^2/sqrt(d*x^2 + c),x, algorithm="giac")
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