Optimal. Leaf size=302 \[ \frac {5 e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}}-\frac {5 e^3 \sqrt {e x} \sqrt {c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac {e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt {c+d x^2}}+\frac {(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {463, 459, 288, 321, 329, 220} \[ -\frac {5 e^3 \sqrt {e x} \sqrt {c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac {5 e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}}+\frac {e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt {c+d x^2}}+\frac {(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 288
Rule 321
Rule 329
Rule 459
Rule 463
Rubi steps
\begin {align*} \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {(e x)^{7/2} \left (-\frac {3}{2} \left (2 a^2 d^2-3 (b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \int \frac {(e x)^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{14 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx}{28 c d^3}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{84 d^4}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{42 d^4}\\ &=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.32, size = 222, normalized size = 0.74 \[ \frac {(e x)^{7/2} \left (\frac {\sqrt {x} \left (-7 a^2 d^2 \left (5 c+7 d x^2\right )+14 a b d \left (15 c^2+21 c d x^2+4 d^2 x^4\right )-\left (b^2 \left (195 c^3+273 c^2 d x^2+52 c d^2 x^4-12 d^3 x^6\right )\right )\right )}{d^4 \left (c+d x^2\right )}+\frac {5 i x \sqrt {\frac {c}{d x^2}+1} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right )\right |-1\right )}{d^4 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{42 x^{7/2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 696, normalized size = 2.30 \[ \frac {\left (24 b^{2} d^{4} x^{7}+112 a b \,d^{4} x^{5}-104 b^{2} c \,d^{3} x^{5}-98 a^{2} d^{4} x^{3}+588 a b c \,d^{3} x^{3}-546 b^{2} c^{2} d^{2} x^{3}+35 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a^{2} d^{3} x^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-210 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a b c \,d^{2} x^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+195 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, b^{2} c^{2} d \,x^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-70 a^{2} c \,d^{3} x +420 a b \,c^{2} d^{2} x -390 b^{2} c^{3} d x +35 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a^{2} c \,d^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-210 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, a b \,c^{2} d \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+195 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \sqrt {-c d}\, b^{2} c^{3} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {e x}\, e^{3}}{84 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________