Optimal. Leaf size=436 \[ -\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{105 b d^3 \sqrt {c+d x^2}}+\frac {\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}-\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}+\frac {\sqrt {c} \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.32, antiderivative size = 436, normalized size of antiderivative = 1.00, 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 = {488, 595, 596,
545, 429, 506, 422} \begin {gather*} -\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 (\text {ArcTan}\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 {\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 (\text {ArcTan}\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 {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 {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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 422
Rule 429
Rule 488
Rule 506
Rule 545
Rule 595
Rule 596
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] Result contains complex when optimal does not.
time = 3.06, size = 306, normalized size = 0.70 \begin {gather*} \frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (45 a^2 d^2+a b d \left (-61 c+45 d x^2\right )+3 b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )-i 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 {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 )+4 i c \left (-12 b^3 c^3+38 a b^2 c^2 d-41 a^2 b c d^2+15 a^3 d^3\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 )}{105 \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 782, normalized size = 1.79 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\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.
[In]
[Out]
________________________________________________________________________________________