Optimal. Leaf size=553 \[ \frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x \sqrt {a+b x^2}}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\sqrt {c} \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b^2 d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.48, antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {488, 595, 596,
545, 429, 506, 422} \begin {gather*} \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{315 d^3}+\frac {c^{3/2} \sqrt {a+b x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b d^{9/2} \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 (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{315 b d^4}-\frac {\sqrt {c} \sqrt {a+b x^2} \left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{315 b^2 d^{9/2} \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 (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right )}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 488
Rule 506
Rule 545
Rule 595
Rule 596
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx &=\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\int \frac {x^4 \sqrt {a+b x^2} \left (-a (5 b c-9 a d)-4 b (2 b c-3 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{9 d}\\ &=-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\int \frac {x^4 \left (a \left (40 b^2 c^2-95 a b c d+63 a^2 d^2\right )+b \left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{63 d^2}\\ &=\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\int \frac {x^2 \left (3 a b c \left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right )+3 b \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{315 b d^3}\\ &=-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\int \frac {3 a b c \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right )+3 b \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{945 b^2 d^4}\\ &=-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {\left (a c \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{315 b d^4}+\frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{315 b d^4}\\ &=\frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x \sqrt {a+b x^2}}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}+\frac {c^{3/2} \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\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 )}{315 b d^{9/2} \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 (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{315 b^2 d^4}\\ &=\frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x \sqrt {a+b x^2}}{315 b^2 d^4 \sqrt {c+d x^2}}-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\sqrt {c} \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\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 )}{315 b^2 d^{9/2} \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 (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\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 )}{315 b d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.32, size = 379, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (5 a^3 d^3+15 a^2 b d^2 \left (-7 c+5 d x^2\right )+a b^2 d \left (156 c^2-115 c d x^2+95 d^2 x^4\right )+b^3 \left (-64 c^3+48 c^2 d x^2-40 c d^2 x^4+35 d^3 x^6\right )\right )+i c \left (-128 b^4 c^4+328 a b^3 c^3 d-243 a^2 b^2 c^2 d^2+25 a^3 b c d^3+10 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (-128 b^4 c^4+392 a b^3 c^3 d-399 a^2 b^2 c^2 d^2+130 a^3 b c d^3+5 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{315 b \sqrt {\frac {b}{a}} d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 1047, normalized size = 1.89 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (a + b x^{2}\right )^{\frac {5}{2}}}{\sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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