Optimal. Leaf size=469 \[ -\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {\frac {c x^2}{a}+1} (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right )}-\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]
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Rubi [A] time = 0.64, antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {958, 719, 419, 933, 168, 538, 537, 424} \[ -\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {a+c x^2} \sqrt {f+g x}}-\frac {2 \sqrt {\frac {c x^2}{a}+1} (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \sqrt {a+c x^2} \sqrt {f+g x} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right )}-\frac {2 \sqrt {-a} g \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}} \]
Antiderivative was successfully verified.
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Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 933
Rule 958
Rubi steps
\begin {align*} \int \frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+c x^2}} \, dx &=\int \left (\frac {g (e f-d g)}{e^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(e f-d g)^2}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {g \sqrt {f+g x}}{e \sqrt {a+c x^2}}\right ) \, dx\\ &=\frac {g \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{e}+\frac {(g (e f-d g)) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^2}+\frac {(e f-d g)^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^2}\\ &=\frac {\left ((e f-d g)^2 \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}} \sqrt {1+\frac {\sqrt {c} x}{\sqrt {-a}}} (d+e x) \sqrt {f+g x}} \, dx}{e^2 \sqrt {a+c x^2}}+\frac {\left (2 a g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 a g (e f-d g) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e^2 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 (e f-d g)^2 \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {f+\frac {\sqrt {-a} g}{\sqrt {c}}-\frac {\sqrt {-a} g x^2}{\sqrt {c}}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^2 \sqrt {a+c x^2}}\\ &=-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {1-\frac {\sqrt {-a} g x^2}{\sqrt {c} \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^2 \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^2 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.08, size = 927, normalized size = 1.98 \[ \frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}} \left (-\frac {\sqrt {a} \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right ) f^2}{i \sqrt {c} d+\sqrt {a} e}+\frac {2 i \sqrt {a} g \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right ) f}{\sqrt {c} e}+\frac {2 \sqrt {a} d g \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right ) f}{\sqrt {a} e^2+i \sqrt {c} d e}-\frac {i \sqrt {a} d g^2 \sqrt {\frac {c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{\sqrt {c} e^2}+\frac {g \sqrt {\frac {g \left (i \sqrt {c} x+\sqrt {a}\right )}{\sqrt {a} g-i \sqrt {c} f}} \left (\sqrt {c} x+i \sqrt {a}\right ) \left (\left (\sqrt {c} f+i \sqrt {a} g\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-i \sqrt {a} g F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c e \sqrt {\frac {g \left (\sqrt {a}-i \sqrt {c} x\right )}{i \sqrt {c} f+\sqrt {a} g}}}-\frac {\sqrt {a} d^2 g^2 \sqrt {\frac {c x^2}{a}+1} \Pi \left (\frac {2 \sqrt {a} e}{i \sqrt {c} d+\sqrt {a} e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {i \sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} g}{i \sqrt {c} f+\sqrt {a} g}\right )}{e^2 \left (i \sqrt {c} d+\sqrt {a} e\right )}\right )}{\sqrt {f+g x} \sqrt {c x^2+a}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 959, normalized size = 2.04 \[ \frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{c f +\sqrt {-a c}\, g}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{-c f +\sqrt {-a c}\, g}}\, \left (-a e \,g^{2} \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+a e \,g^{2} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-c d f g \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+c d f g \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-c e \,f^{2} \EllipticE \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+2 c e \,f^{2} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-c e \,f^{2} \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+\sqrt {-a c}\, d \,g^{2} \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-\sqrt {-a c}\, d \,g^{2} \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )-\sqrt {-a c}\, e f g \EllipticF \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )+\sqrt {-a c}\, e f g \EllipticPi \left (\sqrt {-\frac {\left (g x +f \right ) c}{-c f +\sqrt {-a c}\, g}}, \frac {\left (-c f +\sqrt {-a c}\, g \right ) e}{\left (d g -e f \right ) c}, \sqrt {-\frac {-c f +\sqrt {-a c}\, g}{c f +\sqrt {-a c}\, g}}\right )\right )}{\left (c g \,x^{3}+c f \,x^{2}+a g x +a f \right ) c \,e^{2}} \]
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 \[ \int \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}}\,{d x} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^{3/2}}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\left (f + g x\right )^{\frac {3}{2}}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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